Prisoners are generally poor, so the biggest thing that would affect the prisoner's choice would probably be physical distance to family. This factor may be big enough that the system wouldn't optimize anything else.

Given that the stakes involve being murdered, raped, assaulted by guards, etc. I think many prisoners would accept fewer family visits in exchange for more safety.

Prisoners probably aren't going to get very accurate information about which prison is preferable.

This is definitely wrong, prisons already have strong reputations among criminals, defense attorneys already know a great deal about prison environments from having so many encounters with repeat criminals with inside experience. Given prison choice, defense attorneys would certainly make it their business to know prison reputations to advise their clients.

Even ignoring this, it fails to consider that prisoners may prefer certain prisons for reasons that are harmful to the rest of us; for instance, because the prison has a lot of members of his own gang in it

If true, this would not be different from the current status quo. In California prisons are already basically segregated by gang, and each gang is given yard time separately. This is done to prevent violence.

Prison gangs formed from a kind of arms race of mutual self-defense. Take away the need for self-defense in prison, and people will stop joining them. More to the point, there is really no reason to allow criminals to associate with each other in prison at all. Let them talk to non-prisoners via Skype calls for socialization and send them out on supervised work details if their behavior is good.