Actually, I'd argue people usually choose their enemies, especially rivals. Perhaps not intentionally, but as a consequence of our actions.
Defining such things is a really common trait in lots of different Roleplaying systems. Everything from old school GURPS and Hero to newer games like FATE and Apocalypse World allow players to define enemies.
Edited: Corinthi on 15th Jan, 2016 - 12:38am
Then you're talking about a rival built up from a story and not random. If its going to be based on actions then the Dungeon Master will have to work that into a current character background otherwise a player could insist their character wouldn't have done any action to cause that.
I find the idea interesting. In GURPS 3e it was simulated via a disadvantage that netted the player character points that could be spent to purchase advantages, increase their attributes, add/improve skills; spells; psionics, and the like. A GM could decide not to use it in their game, correct?
I do use this in some of my games. From when a character first starts playing there is a chance they will meet and offend someone in the world. That person they offended can and will use their resources to try to attack or harm the player character in a round about way so that they have to do a lot of digging to find out who they are really up against. Can make some nice in game scenarios as they go from one place to another trying to find who their rival is.
So has the rival not yet been implemented, or simply not enough interest? Nearly a year has gone by since you initially suggested this idea, though looking back not everyone could fully agree on how the process should work.
Myself, I would be more then happy to have the Dungeon Master decide on most of the details though I again offer that it would be more fun to work together with them on it. Some of the best rivals I feel are those that have at least some history with the character, some sort of connection in which to build on.