Hands off! Mine!

Imagine game session in progress.
"You approach a small town in the middle of wastelands." Game Master says. "You are quickly noticed by a lookout on a tower and a  bell is sounded. This startles you as you have not expected people in this area to be on alert. Something's wrong here. What do you do?"
"I make sure my weapon is at hand, but do not draw it yet. I enter the town cautiously, carefully looking out for trouble." John responds slowly, thinking while he speaks.
"And trouble you find." GM responds. "When you reach middle of the town, you see a figure whose face is shaded by the brim of his hat. >>Remember me?<< - He asks. >>Surely, you don't. But I do remember you. How you left me and mom to starve and slave and struggle. But I survived and when she died, I made a vow to take my revenge.<<"

"Hey! That's not in my character's story! You can't do this!"

Mine!

Usually, ownership of game world and characters is very strictly differentiated.
Player can not make any decisions that affect the game world (what is behind the door, is there a convenient niche to hide, is there a distant cousin in this town maybe?).
Game Master is banned from even thinking of affecting the character. Adding a new family member? Be gone, foul fiend! How dare you even hint on something I have not specified! This is MY character!

In a way, this is understandable.
In many cases, Game Master usually has full control over the game world. His word is the law and he is able to shape and control the world however he likes. Players are powerless toys of fate (Game Master), struggling to survive against the odds.

In an understandable reaction, players become overly protective towards their characters and fine tune and polish them almost obsessively. English say "My home is my castle". Players often say the same about their characters.

But... Does it really have to be this way?
Is it really the only way to handle things?
I think not.

Hard work

Game Master usually has many duties to do. I say "duties", because preparing a game session is a hard work. Game Master has to come up with the idea, prepare any items he would like to use (maps, artefacts etc), come up with NPCs that players will meet (and possibly slay)...
GM controls the whole world, runs the story, animates NPCs... In some cases, in the gaming group there is one person who is not really having fun - because he's working too hard to enjoy it.

This type of task separation may be good, especially in setups, where beginner players play with experienced GM. For them, running their own characters and deciding what to do next, can be just about all they can handle.

However, if this setup continues for a longer extent of time, Game Master can experience regular burnout.
And players - if they have that type of mentality - can feel limited by their own characters with no way to escape it.

Tied hands

More experienced players can come up with more interesting and complex characters, whose sheer existence will affect game world - or would, if it was allowed.
Game Master may want to run a campaign that will lead to player characters discovering themselves, but as he can't touch player character...

Sharing

So why not share your powers?
Game Master, let the player say:
"You know, I'm pretty sure that here in this town lives a distant aunt of mine, whom I used to visit when I was a kid. She might still remember me..."
If as a Game Master you allow additions like this, you can later use this NPC - not necessarily to kidnap him/her on the very next session. That NPC can be a source of quests for our players. Plus, suddenly, your team is kind of invested into doing those quests - after all it's family of one of them.
You can encourage players to give such NPCs interesting features. For example, a friendly uncle may have enemies, that will hunt our team some time later. If this is the case, you may discover, that in - so far dull and uninteresting town - you have a blooming criminal world. Or maybe criminals are barely making the ends meet..?
Up to you, really.

Player, maybe it would not hurt to let your GM influence your character a bit?
Sometimes you may not know how to change something or achieve it. Or you may lack idea for something that will make this character stand out from the crowd.
Or maybe you would like to discover things about your in-game self?
Let Game Master add that long lost brother of yours. Or leave some open ends in character story, that Game Master might pick up.

Example

This is something that has happened in actual game.

Character was designed as an orphan, raised by adopted parents, who did not know her "real" parents. When this character is controlled by the player, she has long come to terms with the fact that she will never know who were her parents and she even stopped searching.
GM could have treated it as a finished story (that was initial player's intention), but he could - and did - use that opportunity to add some hidden facts about the character.

He added a story of attempted kidnapping, child getting lost after a desperate saving spell and real parents losing memories in result of the events. Memories that returned years later, but it was too late to find out what has happened to the baby... And those who wanted to get her, were still after her.

The result was a long and very interesting campaign with a very intense culmination.

When it works

For this to work, a trust between players and Game Master is absolutely necessary.
There must be one clear rule: When it comes to the world, it's the GM who has the final say, but when it comes to character, player is the law.
At any point of time, GM or a player can protest and prevent something that would break - be it the world or a character.

But it does give very interesting results and - from my own experience - is worth trying.