"My son (15 now) had a real problem with pain on his "inner elbow" about 3 years ago which caused him to miss an entire season of pitching. He was diagnosed as having "golfer's elbow", medial epicondylitis, to be technical. Golfer's elbow is pretty much the same as "tennis elbow" just on the inside of the elbow."
Medial epicondylitis is actually worse than you describe.
What happened is that he was putting more load on his elbow that it could stand. However, since he was 13 and likely still has an open growth plate, rather than the tendon being damaged, what was damaged (or at least irritated) was the bone to which the tendon attached. The force that was being applied to the elbow was trying to pull the medial epicondyle off of the humerus. Do this enough, and the medial epicondyle will pull off of the bone and have to be surgically reattached.
"The cause is a bit more difficult to pinpoint. After all of this, I believe that it's because of his grip on the ball. I noticed that his thumb had been up the side of the ball as opposed to being directly under the ball and the fingers on top. He ended up putting a "slilder" motion into his pitches. Changing his grip made a big difference in the activity of that muscle and it's affect on the tendon."
It didn't have anything to do with where his thumb was. Instead, the problem was A) how much he was throwing and B) how he was throwing. By how he was throwing, I mean what his wrist was doing.
When they said he was putting a "slider" motion on his pitches, they meant that he was supinating his wrist (turning it clockwise) as he released the ball. This would focus the load on the ligaments of the elbow (and the bones to which they attach). To fix the problem, you have to pronate (turn the wrist counter-clockwise) as you accelerate the arm and release the ball. This shifts some of the load onto the Pronator Teres muscle and some of the bony structure of the elbow.