On a personal level, the mods have a pretty strong preference for random, with I think an option to make a case for a particular swap if there's a character developmenty reason for it, or maybe a "preferences" section where you can say something like, "No one of the opposite sex, that would be too uncomfortable," or "Preferably someone he doesn't know so he can troll gleefully," or "Can I get someone with wacky powers she can experiment with, since she has none of her own?" etc.
It just feels a little sad to me to say "Okay, now everyone is going to swap bodies with their best friend's character!" -- which, if we leave it up to the players, is probably how at least half of the game would choose. A lot of people would go "I love X and Y's interactions, let's swap them!" Which is fun for the players who love their interactions, but isn't exactly a stretch, and is potentially distressing for players who don't know other players well or aren't comfortable seeking out partners. Part of the fun of an event like this is the random, where you can end up swapping with someone your character hasn't interacted with, so they can either wreak havoc with that person's life or form a new relationship, etc.
So: Random is definitely our preference, but we would allow the ability to state preferences, or the ability for a small pool of CR-supported specific swaps.