My son (now a HS sophomore, 16 yrs old) had the same elbow issue as a 12 yr old. We were told to shut him down at least 6-8 weeks, then start a throwing program when he could throw ~20 feet w/o pain (which was at 8 weeks, as I recall). He did some pitching that year after the throwing program, then had similar growth plate-related issues in his shoulder. That was near the end of the summer season, and he stopped baseball and didn't play at all that fall.
The following spring, when my son was 13, his middle school coach planned to use him as a closer to keep his inning count low. In the second game of the season, his elbow started bothering him again. Another 8-week shutdown. This time he did physical therapy (which had not been recommended when he was 12). That summer and fall he played 1B and closer--didn't have any problems (though his coaches were pretty careful with him).
Since that time, my son has been a starter throwing a moderate number of innings for school ball in spring, then summer and fall travel teams. No problems for 2 years now (fingers crossed). He seems to be done or very nearly done growing.
The problem years are during major growth spurts. (My son did his fastest growing at 12 and 13.) My advice is to find a physical therapist who has experience working with young pitchers. Have the therapist evaluate and treat any imbalance or flexibility issues, and also get your boy in the habit of doing regular PT at home for maintenance. Buy some 3 lb. dumb bells, therabands or whatever equipment your therapist recommends, put the gear where the boy can see it, and do a little nagging until PT becomes a habit.
It's frustrating, but the initial pain at the inside elbow is his body warning him to stop--make sure he knows he should pay attention. I saw a kid on my son's team play through elbow pain at 3B until his UCL pulled the growth plate away from the bone. That required surgery and full year to recover.
A couple of sports medicine docs told us that the issue most commonly shows up in kids who throw hard / are strong for their age--which made my son feel a little better. Missing most of two years of pitching didn't seem to hold my son back much--he should be pitching on varsity as a soph this spring.
I know it can be hard to keep a kid from getting frustrated and either quitting baseball or trying to throw too soon. I told my son many times that if he wanted to stop pitching, I'd support that; but that if wanted to keep trying I would drive him to PT, see any doctors he needed, etc. as long as I could afford it and he was willing to work at it (but only if he put in the work). There were days I thought he was going to give up, but he stuck it out.
Good luck to you and your son. (And sorry this is so long.)