Both Sides Have a Point

Avram: (gestures at Perchik and Mordcha) He's right, and he's right? They can't both be right.
Tevye: You know... you are also right.

—Fiddler on the Roof

Alice is faced with two different opinions: Bob strongly believes in one thing, and Charlie in another. The easiest choice would be to simply pick a side—decide that Bob is right or that Charlie is right. But Alice won't do that. The second easiest choice would be to simply remain neutral and urge them to Agree to Disagree. But Alice won't do that either. And she will neither pretend that the two opposing views are actually the same thing, nor conclude that it's merely a matter of perspective. Finally, she will not engage in some extreme mental acrobatics, simultaneously but separately agreeing with both opposing views.

All that is left for her, then, is something much harder: to try her best to see both sides fairly, and value the merits of each side's arguments.

In Real Life, this process can be very stimulating and rewarding, and it is also necessary for people to truly coexist in a a decent manner. In fiction, it can serve to enrich the morality of the setting and avert Black and White Morality.

This trope might lead to an Author Tract unless it's Played for Drama - focusing on Alice's emotional reactions to the dilemma rather than the dilemma itself. When Played for Laughs, it often strays even further from the actual issue.

Contrast What Is Evil?, which is an aversion of this trope: The villain tries to invoke Both Sides Have a Point, but it is made clear to the audience that he does not, in fact, have any valid point whatsoever and the protagonist is also very unlikely to listen. This aversion is much simpler than playing the trope straight, and is thus far more common - especially in action stories where the audience wants to see big fights and will likely find a valid moral debate to be a boring disruption. Also contrast Culture Justifies Anything, where it's very likely that at least one side does in fact not have any valid point.

Not to be confused with Double Weapon, where both sides of your weapon have a point. Compare Grey and Grey Morality, Black and Grey Morality and White and Grey Morality as well as Rousseau Was Right and Good Versus Good. Characters stuck in this situation may decide to Take a Third Option. Beware of falling into the Golden Mean Fallacy, where a compromise is reached, but one side is flat-out wrong, and has no valid point after all.

Examples of Both Sides Have a Point include:

Anime and Manga

In Karakuridouji Ultimo, the protagonist Yamato learns he is the cause of an apocalypse in the near future. (This is part of the premise, so it's not really a spoiler.) Yamato chooses to avert this by finding every person in the world who would be involved in the event and understanding their points of view so that he can choose the best possible action once the time comes.
A lot of Hayao Miyazaki's films are based on this kind of premise. He dislikes the limiting assumptions of a lot of conventional media that evil exists and must be defeated by good.
- Princess Mononoke is perhaps the best example of this, with every character having a reasonable explanation and motivation for their actions. San is harsh and violent - but only wants to protect her home, family and the natural world. Eboshi wants to kill the god of the forest - but is a benevolent leader, good to her people and kind to lepers and you can fully see why her people are willing to die for her. The protagonist Ashitaka is completely neutral and genuinely wants the best for everyone. Even if the consequences of their choices are ultimately negative, you can see why they did it.

Fan Works

Renegade: Word of God is that the conflict between the Global Defense Initiative, the Brotherhood of Nod, and the Citadel is a variation of this. Specifically, that no one is really right, but everyone is wrong on certain points, which is what leads to their conflicts.

Film

The entire film Fiddler on the Roof runs on this trope. Tevye is caught in the clash between the traditional world and the modern world. He's a really smart guy, but poor and uneducated. He tries his best to be fair and see both sides of the situation, with many inner monologues about "on the one hand [...] but on the other hand". In the page quote above he gets ridiculed for not simply picking a side when two guys who both have valid ideas stick to parroting slogans at each other instead of making more nuanced arguments for their causes.
The Social Network is done this way, and the characters themselves reach this conclusion: None of them is truly unsympathetic, and they all have more or less valid claims and complaints.
In Team America: World Police, both 'dicks' and 'pussies' have a point, according to Gary's (plagiarized) speech at the end. The 'assholes' on the other hand, just make everything worse for everyone.
The first X-Men 1 film had this trope for the political hearing which Jean Grey debated with politicians concerning mutants. Both sides brought up good points which was the intentions of the director.

Jokes

The trope is reduced to absurdity in an old Jewish joke. Two Jews come to a rabbi to resolve a dispute and present their arguments; they also bring along a witness. The rabbi, after leafing through the Talmud for a couple of hours, finally says: "Shlomo, you are right. But, Moyshe, you are right as well". The puzzled witness asks: "But, rabbi, how can two men with completely different opinions be right at the same time? It's impossible!". The rabbi replies: "You know, Joshua, it turns that you are right as well!"

Literature

Felsic Current
The Sheriff of Nottingham is able to do this to himself in In A Dark Wood, Michael Cadnum's White and White Morality retelling of Robin Hood. Halfway through the book, he is able to recognize that although Robin Hood is an outlaw, he is also a good man. It isn't until the end of the book that he is able to find a point of reconciliation between this and his duty to uphold the law.
The Audacity of Hope by Barack Obama use this trope as a cornerstone for much of it's portrayal of the political landscape.

Tabletop Games

The later versions of Mage: The Ascension used this perspective. The Technocratic Union wants a stable and democratic reality where everyone is able to create miracles through technology. They have largely succeeded: the modern world with computers, airplanes and modern medicine exists by their design. Their opponents, the Traditions, prefer a more unstable (ahem, dynamic) reality with more personal freedom - a freedom of expression that includes rewriting reality itself rather then merely writing words. (The original version had this same conflict of interest, but hardcoded that the Technocracy's ideals made them Dirty Commies.)

Video Games

In The Amazon Trail 2, one location has you talk to an oil executive and a native from the area. The executive wants to drill for oil, and the native doesn't want the environment to be spoiled. Now while the game has a bias to the native, the game will only reward you if you listen to both characters about the issue.
This is the Paragon resolution of post-loyalty mission conflicts between members of The Squad in Mass Effect 2.
The Templar/Mage conflict in Dragon Age II is the epitome of this trope. The Mages are horribly oppressed by the Chantry's Templars, imprisoning them to keep the city safe and treating all Mages as dangers. At the same time there are a lot of Mages who seem to turn to Blood Magic and the like, due to the weakness of the Veil in the area. Better safe than sorry?
This is what drives a lot of the Grey and Gray Morality in the Geneforge series, with even the more "evil" factions such as the Takers or Barzites making the occasional valid point.
The civil war subplot in Skyrim is all over this: the Stormcloaks are correct in that their traditional religion is being unjustly oppressed, the Empire caved in in order to end the Great War, and that it's become a decrepit, corrupt entity. The Imperials are correct in that the Stormcloaks are full of xenophobic assholes, Ulfric used a traditional excuse to justify murder and attempt to seize power, and that if the Empire starts to come apart it will be easy pickings for their enemies.
There's something of a Deconstruction in KotOR II. Inside Ludo Kressh's tomb, the player is faced with a series of illusions. In one of these, the player's companions are about to attack Kreia, (the player's mentor) and the player must decide who to side with. However, if the player answers "I won't attack you, but I won't stop the others from attacking you either", Kreia exasperatedly scolds you and everyone present tells you that "apathy is death".
If you were to join the chaos side in Shin Megami Tensei, you might be surprised to find out that they wish to free the world from tyranny. Angels aren't wrong to consider demons to be evil though. After all, they certainly can be evil. Angels aren't necessarily saints, however. They can indeed be oppressive. But nevertheless, the player might want to choose the neutral path and decide to remain with humanity. Regardless of what option you choose, one of your friends will agree with your decision.

Web Comics

In Dinosaur Comics strip 635, T-Rex is asked what God thinks about intelligent design:

God: I LIKE TO DANCE IN MY UNDERPANTS T-REX
T-Rex: He says - there's some merit to both sides of the issue?

Western Animation

Done on The Simpsons in the episode The PTA Disbands when Principal Skinner and Mrs. Krabappel are trying to convince the parents at a PTA meeting of their respective positions. Mrs. Krabappel argues that Skinner's budget cuts are harming the education the parents' children are receiving, and that they need the resources to do their job. The parents are inclined to agree with her until Skinner points out that the school is on a very tight budget as it is, and for the school administration to get what the teachers are asking for they'd have to raise the parents' taxes. That gets the parents complaining about taxes being high enough as it is, and the debate between Skinner's and Krabappel's positions ends up going back and forth. The episode ends by Skinner and Krabappel deciding to Take a Third Option and rent out the school's cloakrooms to the prison system to raise extra money, although the writers don't provide an answer to the taxes vs. education quality debate.
South Park uses the Golden Mean Fallacy a lot to find a middleground between two opposing sides, ultimately arguing that each side is partially correct.
The animated TV adaptation of The Lorax does acknowledge that a lot of people would lose their jobs if the Thneed factory shut down.

Real Life

This is the reason some people prefer to use the Golden Mean Fallacy when considering controversial topics like "Which political party really is worse for the country" or "Evolution vs. Creationism" and so on.
It's also why political power can and frequently does swing back and forth between different parties in democratic countries, as voters decide they like one party's policies at one time and then decide to switch to another party's policies later on. Sometimes parties who win elections and form governments end up plagiarizing parts of their opponents' platforms in order to broaden their own appeal in the electorate.
- Similarly, in the Canadian 2011 election, the arguably most centered party (Liberal) was squashed out in favour of the NDP, which is, for the most part, more to the left, and the Conservative party, the right-most major party, which was already strong beforehand. It's up to debate how much of this was because of Jack Layton, and how much it was because of the poor showing of the Liberals (including the fact that they didn't claim Both Sides Have a Point).
  - Considering that voting returned to the status quo in the next election when Jack Layton was no longer leader of the NDP, this was because of Jack Layton.