Placebo Eureka Moment

Alice has a problem and she goes to Bob for some advice. She pours her problems out to him, unaware that he isn't paying attention, but by the end of their "conversation" she's figured out the answer to her problem and thanks Bob for his help. On the other hand, Bob may be all set to give her some advice, but before he can get a word in, Alice comes up with a solution and runs off, thanking him, and leaving Bob bewildered, though he'll still probably say, "You're welcome."

In short, this is when a character has a Eureka Moment without being inspired by the other character, but acts like the other character gave them just what they needed. Sometimes the other character isn't actually listening in the first place - they're asleep, or they've left the room, etc. Sometimes it's an animal, someone who doesn't speak English, or The Silent Bob. A common variation is for a character to be visiting their therapist, who doesn't say much more than "hmm," and "I see," yet they leave the session feeling much better (in fact, psychoanalysis actually recommends that the therapist say as little as possibly, partly to prevent them from putting words in the patient's mouth and partly to encourage the patient to talk themselves).

Also known as Rubber Ducking IRL.

Film
"Rafiki: My work here is done."
 * Charlie and The Chocolate Factory (the second movie) - "Oh, you're good."
 * Airplane II: The Sequel has this with Ted Stryker, in a flashback.
 * In The Lion King One and A Half, this happens the second time Timon meets Rafiki.

Literature

 * Some of Hermione's I-think-I-just-had-an-epiphany-I'm-off-to-the-library moments in Harry Potter were like this. Lampshaded by Ron at least once (he hates it when she figures something out and then leaves without telling them what it was).
 * Rin, in Shannon Hale's Forest Born has the gift for helping people this way, due to her quiet and accepting nature.
 * Momo, by Michael Ende. Momo is a little girl who basically solves everyone's problems by simply listening to them. She never gives any advice at all, but the people who talk to her suddenly come up with ideas and solutions.

Live Action TV

 * Black Books, Bernard and Manny visit the same psychiatrist, and both of them experience (temporary) epiphanies. The shrink never says a thing, and gets showered with money for her trouble.
 * Happens in the Castle episode "Punked". Alexis asks her dad how to know if she's in love, but before he can give her any advice, she rambles on about it, coming to her own conclusion, but thanking him as if he gave her the answer. Quoth Castle as she runs off, "Glad we had this talk."
 * Frasier, when Frasier asks for his brother's advice, he basically uses him as a Straw Therapist while he diagnoses himself.
 * This sometimes happens with Charlie's interactions with his therapist in Two and A Half Men. Sometimes he tries to make it happen in order to reach his preferred conclusion, usually with little success.
 * Monk, as well, at least once each with Dr. Kroger and Dr. Bell, as well as a time or two where he used a non-therapist as a wall off of which to bounce ideas.
 * Partially averted on Bones when Angela goes to Dr. Sweets for advice on how to deal with Hodgens now that she and he had broken up. Sweets recommended a full therapy regimen (starting with two sessions per week) but she pronounced herself "fixed" after talking with him for 2 minutes, with her doing most of the talking.
 * The therapist Maddie talks to (or rather at) in the Jonathan Creek episode "The Scented Room".
 * The second episode of Scrubs did this, where first J.D. and then Elliot did this with Dr Cox. This cemented Dr. Cox as the guy people (especially J.D.) came to with problems, much to his annoyance (especially when it involved J.D.).
 * With the Doctor around, this is almost given. He'll be talking to his companion and then he'll explain what he had figured out. His companions will still wonder what he says!
 * House, usually at least Once an Episode. Sometimes House and Wilson will be in the same room and talking, but having two entirely different conversations, and something Wilson says or does causes House to get up and leave, generally without explanation.

Theatre

 * In Once Upon a Mattress, the Queen asks the wizard to help her think of a test to give Princess Winifred, but ends up just complaining without letting him get a word in edgewise. She ends up coming up with the idea all on her own, but still calls the wizard "a genius" for it.

Video Games

 * The therapist in Phantasmagoria 2 had surprisingly little to say about Curtis's various revelations: everything from being dressed as a girl by his horrifically abusive mother, to his attraction to both his female co-worker and male best friend, to his interest in bondage. You'd think there would've at least been a comment of note. Oh well, Curtis seemed pleased with the sessions.

Web Comics
"Frank: Wait. I arrived at my conclusion just by thinking about it.
 * Early In Order of the Stick, the ghost of Roy's father made a habit of appearing to him when he tried to get to sleep. After briefly forsaking the rest of the party, he didn't appear, but Roy still tried to talk to him anyway, leading to this trope.
 * In one Arthur, King of Time and Space strip, Merlin discusses something with Arthur over breakfast. Merlin then admits he just needed to say it out loud, and Not a Morning Person Arthur replies that obviously Merlin knows him better than to actually ask his opinion at that time of day.
 * In Wapsi Square, Monica thanked Darren for his advice without realizing he never said anything.
 * Used in this Kevin and Kell strip, where Frank seeks guidance from his advisor Simcha.

Simcha: Sometimes this is actually a pretty easy gig for me."

Web Original

 * In Legendary Frog's flash video "Kerri's Big Invention", Kerrigan asks a monkey for advice, which responds to her in generic monkey noises. Kerrigan admits that she has no idea what the monkey said, but is now somehow inspired. She then immediately gets back to tinkering.

Western Animation

 * In the South Park episode "Are You There, God? It's Me, Jesus," Kyle asks his baby brother Ike for advice and just gets gibberish in response, but he comes to a conclusion on his own.
 * In TinkerBell and the Lost Treasure, Terence vents his frustations with Tink to an owl, and actually answers the owl's hoots ("Who?"), working through his problem this way. After he leaves, Fairy Gary shows up, looking for advice about the guys making fun of his kilt.
 * Kung Fu Panda Holiday Special has this one where Po thinks of a way to make sure he and his dad spend time together during the Winter Festival, so he mumbles in front of the guard who remained stoic. He figures out a solution regardless and thanks the guard who still remained stoic.
 * Happens in The Simpsons episode "Who Shot Mr. Burns Part 2" when Lisa finds a clue and then thanks a nearby pigeon for pointing it out to her.
 * Occurs in Play It Again, Charlie Brown between Lucy and Charlie Brown.

Real Life

 * There is an old academic advice that basically says, if you want to understand what you're learning, find someone who knows even less than you to explain it to. In other words, in the process of breaking down an idea for someone else to understand you also get a better understanding of it (this is very much a truth; even if you thought you got it before, it can reveal a lot of mistakes or parts that you have skipped.)
 * In software development, "teddy bear debugging" is a well-known technique where you try to fix a problem by describing it to a coworker, friend, or even a stuffed animal. The mere act of putting it to words often allows you to figure it out on your own.
 * The ELIZA program, a "psychoanalytic" computer program where people would type in their problems (for instance, "People don't like me") and it would just throw their statements back at them in the form of questions ("Why do you think people don't like you?"). The majority of users said the program helped them feel better.