I'm sure you all know this, but I am a huge proponent of character-driven games. As a GM, I love it when the players start throwing ideas and hooks at me.
(Side note: I need to work on being better at this as a player)
Players contribute to the story in a number of ways, some of which are less obvious than others, and their first contribution for the vast majority of games is their character, because the characters are what the GM should structure the world around.
This is one of the ways in which I prefer characters who require construction. Much as I have enjoyed all of the versions of Dungeons & Dragons over the years, it wasn't a system that was structured towards that sort of "GM and players telling a story" that I like. First and second editions were especially bad at that, because every fighter was (essentially) just like every other fighter. The differences were all in your skills (and weapon proficiencies), but that was only as useful as your DM made it. And the difference between WP: Khopesh and WP: Broadsword only rarely came up.
I can't speak to fifth edition, however. I suspect that the "backgrounds" that it uses provide some of those story hooks which I so desperately crave as a DM.
But most games with constructed characters allow players to take advantages and disadvantages. You know. One eye. Cursed. True Love. Higher Calling. Things that either constrain your character in some way so that you can get more points to boost your character somewhere else or that boost your character's abilities in some way.
Realistically, these are almost all story hooks. Even the advantages can provide good hooks for the GM to use. Contacts and Allies? Who are they? Cursed? By whom and why? Dark Secret? What is it?
Skill selection can provide some of these story hooks, but often that requires deeper inquiry from the GM to the player.
As a player, I'm generally rotten about giving my GM good explicit hooks. I'll freely admit that. I'm a big fan of the amnesiac orphan characters in D&D-style games (which some GMs really appreciate and some really despise). 13th Age kept me from doing that via its backgrounds that invested me in a bit of worldbuilding (I'm pretty sure Wade is still shaking his head a bit at the presence of an Imperial Inquisition in his world). But I really love interesting advantages and disadvantages.
As a GM, I love to see players like me when it comes to advantages and disadvantages, because it gives me ideas. Even players who aren't excited about a game can steer it quite a bit with their selections.
In my current Legend of the Five Rings game, for example, I have one player who didn't choose any advantages or disadvantages. He has one disadvantage, but it came from his clan's Heritage roll and not from his choice. As a GM, this has made it difficult for me to hang any kind of story off of that character.
With the Dungeons & Dragons game nearing its end (within a year or two, I suspect), there will almost certainly be another game spinning up to take its place. I need to be sure to give my GM as many plot hooks as I can ...