I'd recommend you contact some of the MLB club's trainers and ask about who they use, or contact the pitching coaches at some of the more prominent college baseball programs in your region and ask who they refer their players to for arm injuries. If you go to a surgeon who has plenty of experience and a successful track record, your son will very likely be just fine, provided he does everything he's supposed to in his rehab.
Personally, there is no way I'd let my son have this procedure done by someone with only a dozen or so Tommy John procedures under his belt. When my son suffered this injury and needed TJ surgery, we were referred by our medical group to a number of surgeons, none of whom we felt comfortable with. The first few either had never done the procedure or had only done it in very limited numbers, but they all felt qualified and capable of doing it (can you say guinea pig?). I was also hearing from them that the success rate was only 50-80%, depending on which doctor we were talking to. We finally were referred to a surgeon who'd done about 25 Tommy John surgeries, and the first thing she told me was that she was NOT qualified to work on a baseball player. She performed these largely on accident victims or other people who'd suffered UCL tears and simply needed to be able to lead a normal life. She was very up-front in telling me that she didn't consider herself experienced enough to operate on a baseball player, if he had to get back on the diamond (why else would you do it? I wondered). She referred us to the head of sports medicine for their university medical center, a doctor who'd done the procedure more times and who is the team doc for a very high level D1 program out here on the West Coast. After talking with him and hearing how he felt the procedure was a bit difficult to get just right, we were very concerned. While we felt he could probably do it, we weren't completely comfortable since he talked about how difficult it was; and he felt a good success rate was in the 80-85% range.
All the time, there was a surgeon we wanted to do the procedure, someone who is the team orthopedic surgeon for an MLB club and their affiliates, but our insurance wouldn't cover him since he wasn't contracted with our primary care physician's medical group.
Finally, a scout that I know asked me bluntly "If Dr. A is willing to operate on your son, why are you even considering anyone else? Don't be an idiot, step up and get him fixed by one of the best." That hit home pretty squarely, and we went with the guy we wanted all along. I had to pay cash (still fighting with our insurance to try for some reimbursement), but I've never once regretted having his TJ surgery done by the right guy. Son's rehab has gone very well, he has full range of motion and is throwing well without pain. In September he'll be joining a local jr. college, competing to start at catcher. In the meantime, he's been throwing a lot,is working on getting his hitting stroke back and also helping a AAA team out as a volunteer bullpen catcher when they come to town to play our local AAA team. He's been having a ball catching those pitchers, and it's helping him get ready for the day he steps back on the diamond in a few months.
Get the right guy, someone whom you're comfortable with, and who you have confidence in. That way, you'll know you gave your son his best shot, regardless of the outcome. If you're worried about the cost if you have to pay for it, it's about what a deluxe family vacation to Hawai'i or the Carribbean would cost. It's even less expensive if you consider the cost to go on your own to be the difference between paying for all of it, and what you'd pay to go with the surgeon your insurance will approve. Either way, you're going to pay some money. Figure out if the extra cost to get the right guy is worth it for you. It sure was for us.