So, I know this thread is old, but wanted to comment on it for those that may be looking for recent information.
My son was diagnosed with tricep tendinitis a few years ago (when he was 13 and he is almost 16). He has had a few rounds of it on and off over the past 3 years. Usually shutting down for 2-3 weeks, RICE and stretches/bands works it out and he's fine. Just the end of this June, he said he was having the issue again but it went away after some rest and such. Fast forward to the end of July and it was still bothering him and he noticed his velo and mechanics were off. Went back to his ortho who did all the elbow tests and shoulder tests and determined it was actually coming from his shoulder. Diagnosed him with rotator cuff tendinitis. Explained the whole science of how the rotator cuff and surrounding shoulder muscles/tendons can cause pain in the elbow and explained the dynamics of pitching and the shoulder. I found that very interesting! Since then, he has been given specific strength and stretching exercises to do. He's in PT doing some other modalities also and will be back in his gym 4X a week to get the shoulder and supporting areas strong and flexible. Oh, he's going to start doing yoga once a week also for the much needed flexibility!
But, I do recommend, if you son has any elbow issues, or notices his velo and/or mechanics are "off", find a good ortho and get it checked out. Also, implement a solid strength and stretching program year round.