Hey there, and welcome to Pit Bull Forum...
So, I am sure you know, that Pit Bulls are not exactly the "high demand" dog to adopt; there are more in the shelters and rescues than there are homes for. First, if there is any way that your son can alter his lifestyle or make other changes that would allow him to properly take care of the pup, that would be the best answer for everyone concerned. I am sure that you have throught of this already, though....
Second.... here is what I suggest; do a search on
www.pbrc.net and
www.petfinder.com (and ask here) to find the pit bull rescue group closest to you... contact them and explain your situation clearly. Then tell them that you/your son or whoever is willing to commit to the responsibility is willing to help do the foot work to put her into the home. Let them know that you will continue to hold the pup as long as you are able (ideally until a home is found), that you will cover the cost of a spay, have vaccinations up to date, have her treated with Advantage for fleas, and have her treated for worms. (These are all expenses that rescues put into each of their dogs, and rescues are almost never "funded".) Let them know that you will supply them with a complete and fun bio of her to put on line, and e-mail pictures. If they want to meet her, make sure you go to the effort of transporting her to the rescue... These somewhat simple things will take a huge load off of the rescue, and will increase the chance that the will help place the pup. Tell them that you would like for them to receive interested people, do their application and/or screening process in hopes of finding a quailified forever home.
Sorry to be long winded... and i hope that everything works out for you, your son, and the pup.