Match Maker Quest

The Player Character has overcome incredible odds, defeated mighty enemies, righted wrongs and punished the wicked, and even fetched the [Blue Book] from the top shelf! But none prepared him/her for... the Match Maker Quest!

An NPC (or three) are down on their luck in the love department, and can't seem to get that special someone to notice them. They're so desperate and mopey and (a)pathetically resigned to their doomed loveless future that they will spill their guts about these troubles to the first stranger to cross their paths (That means you). Rather than find a solution themselves, they will ask you to do something for them, usually involving a Fetch Quest for some engagement gift, Impossible Task, or plain old bling to impress their potential partner. Other times, it suffices for you to use your prodigious dialogue screen channeled charisma to convince their intended to love them back. Occasionally, the match is already made and the quest centers on solving some other difficulty, like getting the parents to let them marry, gathering wedding materials, a dowry, or helping them set up a life for themselves.

Depending on the game, you may find yourself in this situation so often you practically become The Matchmaker and end up fixing everyone's love life. If you're lucky, the game developers won't forget to give the Player Character his/her own Romance Sidequest. Of course, some developers like to play with this trope and include options for you to sabotage this burgeoning love or outright steal them from the quest giver.

The Match Maker Quest is usually a necessary middle link in a Chain of Deals, and almost always rewards you by dropping some unique love themed loot. Hey, who cares if the +5 Vorpal Sword is pink and fills the air with Heart Symbols? Headless bodies won't complain! Isn't it amazing how RPG's manage to make even love reward you with implements of violence?

See also the closely related, more generalised trope Sidequest Sidestory.

Action Adventure

 * In The Legend of Zelda the Wind Waker, you have to help two painfully shy NPCs (who are too shy to even say hello to each other, even though they see each other every day) get together. This involves taking and showing them pictures of each other.
 * Spirit Tracks has one that has become quite infamous due to what happens after said quest.
 * Majora's Mask has the very long Kafei and Anju quest, where it appears that Kafei has gone missing just before his marriage with Anju. It's not as simple as it looks, and you can't even do this quest until very deep into the main game.
 * Skyward Sword has one too. It's played with though: firstly, you have two possible options, you can either give Cawlin's love letter to the object of his affection, or you can end up giving it to the hand in the toilet. Secondly,  Both paths give you the same reward, so you don't need to worry about Lost Forever.

Adventure Game

 * In Quest for Glory IV, one of the quests is to get a feuding married couple back together by making them realize how much they miss each other. However, neither of them will ask you to do so, there's no reward for doing so, and it's completely optional.
 * This is pretty much your main goal in Hamtaro: Ham-Ham Heartbreak.

Dating Sim

 * In May of the 2nd year of Tokimeki Memorial 2, one of the protagonists' best friends, Junichiro Hokari, will phone him to confess he's in love with one of the girls of the cast. You, as the protagonist, can, with the help of your childhood friend Hikari, can then set up a double date between you, Hikari, Jun, and Jun's love interest, in order to help him get close to her. Of course, if the girl Jun targets is the same as yours, this is the clue for restarting your game, unless you feel really confident in your combat stats...

Role-Playing Game

 * In the game Dragon Age, the player has the option to complete the quest "Cammen's Lament" where they must help a young Dalish elf named Cammen woo his love, Gheyna, who refuses to marry him because he is yet to become a full-fledged hunter.
 * The player also has the option of being an absolute shit and breaking them up forever, potentially by seducing one of them. Doing so has no benefit beyond sadistic pleasure.
 * May not be entirely an example since you're helping out a party member instead of an NPC this time, but in his character quest, Oghren asks for the PC's help in getting an old flame of his interested in him again.
 * In Dragon Age II, Aveline's sidequest involves her attempting to court a fellow member of the city guards. Depending on whether you like plots like this, it's either absolutely hilarious or excruciatingly painful to observe. Or anything in-between.
 * In Neverwinter Nights Hordes of the Underdark, you meet an angel who managed to learn where he would meet his one true love, without learning her name. You have the option of asking the Knower of Names for the name of the angel's one true love, it is randomly selected between Aribeth, Nathyrra, and the player (if female).
 * In Fable, you can bring together a boy and a girl who previously met at a party, by bringing his love letter to her(or claim that the letter is yours, and get the girl.)
 * In Fable II, the player can find a date for a farmer's son. The farmer would like you to set him up with a nice girl, but he would prefer a nice boy.
 * Fable II also has another event that either inverts or subverts this depending on what you do. There is a ghost who was left by their fiance at the altar and in revenge they enlist you to woo their (still living) ex and then crush their heart so they'll share the same pain they did. You can either choose to follow up with this or to derail the would-be revenge plot by actually falling in love with said ex. Either way,.
 * In Fallout 3, Angela Staley wants to catch the eye of Diego. However, he's an acolyte and doesn't really have the option of wooing her AND staying in the church. There are three options here:
 * Threaten to tell the priest. Diego will confess to the priest and break the relationship off to stay in the church.
 * Lie to the priest that his acolyte slept with a woman. Diego will be excommunicated and gets the girl but no marriage since he's living in sin.
 * Force the issue by giving Angela ant queen pheromones to seduce him into quitting the church and marrying her. If you sit through the wedding, you get positive karma.
 * Also, Ronald Laren would like The Nuka-Cola Challenge to be this, but it doesn't work out that way. As in, you have the option of giving him the dough after fetching it so that he can use it to take advantage of her naiveté but this nets you bad karma, even though he doesn't succeed because the target is too stupid to understand his advances. If you want to be evil and have a female character, you can tell him to fetch the dough himself by proposing a threesome; he rushes off eagerly, only to get killed.
 * And you don't get negative Karma.
 * Fallout: New Vegas has Jack and Janet. Jack lives inside the Boomer's air force base and is afraid to leave but looks out with binoculars, Janet lives outside the compound and often looks in with binoculars but cannot enter because the Boomers attack anyone who approaches with artillery out of paranoia. To finish the quest you need to convince the Boomer Elder to allow her to enter the base and join the Boomers, then talk to Janet's boss and get her out of her contract with the Crimson Caravan Company (preferably without giving up her pay). If you want, you can lie to Janet that she's clear to go without clearing things with the Boomers, upon which she'll be obliterated by artillery shelling.
 * In Knights of the Old Republic, there is a Romeo and Juliet plot on Dantooine, that the player character can turn into a massacre, or a happy ending.
 * In Tales of Phantasia, if you do one of these quests near the beginning, you can meet the one of couple's descendants in the future.
 * Tales of the Abyss has a sidequest near the middle of the storyline where you can match up a pair of generals on the opposite sides of a war (naturally, the quest can't be completed until after you stop said war).
 * A somewhat roundabout one in Dragon Quest VIII winds up a subversion. In order to be properly recognized as a man and marry Princess Medea, Prince Charmless needs to obtain an Argon Heart. He's finally talked into getting it when his father describes Medea's... charms, but needs your help. He spends the entire mission bumbling around, talking down to your party, making unreasonable demands, trying to ride the pretty horse that is actually Medea under a Baleful Polymorph Curse, and winds up
 * In Jade Empire, you come across an engaged couple having trouble because the man made a Childhood Marriage Promise to a local gang leader, who intends to hold him to it. If you can convince her to let him go, you get a follow-up quest to find a prospective mate for her.
 * Mass Effect 2 has an asari and a krogan in a relationship, but she's having doubts now that he's starting to take it seriously. You can convince her to continue the relationship or break it off.
 * If you convince her to continue the relationship, you encounter the couple again on Tuchanka, the krogan homeworld. You overhear the krogan trying to explain to his asari girlfriend why living on a barren, barely habitable krogan wasteland is so much better than living on an ultra-modern asari-ruled planet...
 * And then, in the third game,
 * In Final Fantasy X-2, one side quest involves playing matchmaker for two dozen monkeys infesting a tourist trap the PCs want to shut down. Another involves telling basically every woman in the world that an uncharismatic but wealthy corporate heir is available.
 * There are several of these in Morrowind. The first involves a woman who has fallen for the bandit who robbed her, and wants help tracking him down. Surprisingly, it turns out well. Another in the Tribunal expansion involves helping a woman who is far too busy with her work to meet men find a husband.
 * Skyrim has the quest The Book of Love where the player helps out 3 couples, including, get together.
 * In Final Fantasy XII, you can help a viera in Rabanastre to meet her 'soul mate' - either a Seeq, or the hume in the plaza.
 * This happens a few times in Xenoblade. One of the more memorable ones is also one of the earliest: there's a love triangle, and the protagonist has to decide which pairing to support by finding a love potion (or equivalent) and giving it to the right person. Problem is, the main character has the ability to see the future, and both options end badly. There's no way to Take a Third Option, either - unless you simply ignore the quest. Oh well, it's good XP...

Simulation Game

 * Many a Harvest Moon fan have done this, due to the fact that the games have multiple romance rivals.

Turn-Based Strategy

 * A few quests in Final Fantasy Tactics A2 technically are of this sort. Technically, because the game isn't really designed to handle running around talking to NPCs, so it's usually just an excuse for killing things.

Wide Open Sandbox

 * There are hints of this in Bully, as some students ask Jimmy to put presents in other people's lockers, but they are only minor sidequests.