Backstory

Take Inspiration from Players
I like to insist that players give some thought to character backstory, especially if they reach 2nd level (or the equivalent in non-level systems). By that time the character has been played enough that even though the PC may have been created randomly, with no "concept" in mind, the player has enough experience with the PC to start to get a feel for "who they are". PC death is a very real part of most of the games I referee, so getting attached to a PC or spending too much time on developing an elaborate backstory can be a delicate balancing act. The potential benefits outweigh the risks, however.
One of the main reasons I like to hear about PC backstory from players is that as referee I draw inspiration from the backstories. I like to improvise and build as I go, so anything the players give me that catches my fancy or prompts a thought, I like to jot down and make use of during play. It makes coming up with ideas more of a shared task and I think also makes for a better game where all involved have more ownership. I have occasionally met some resistance from players who prefer everything be thought out ahead of time by the referee and for them I may be a poor referee. Many players respond positively, however, when they discover I am looking to them for ideas we can use to further our collective adventure in role-playing.
Another reason I like to hear about their PC's backstory is that it tells me what the players may want to see and do in game play. It gives me ways to draw them into the story and make it more personal to their characters.  I like to play in games that as a player I have some input and naturally like to referee in a manner that does the same. There are narrative systems and indie games that have system mechanics devoted to backstory and player involvement in shaping world and story, but I have found something very similar can be accomplished with conversation and attention to those aspects during play. Much of what goes on during a role-playing game has little to do with the actual rules.