Dr. Julian Sanders wrote Dodgy Elbows and Dodgy Elbows Revisited which cover elbow tendinosis in greater detail and scope than I am willing or even qualified to do here.
To summarize very briefly, the pain is caused by muscles which, while they connect and hurt in the elbow, are actually controlling the wrist and hand, so we primarily fix this with exercises of flexion, pronation, and supination of the wrist (TODO: add pictures). Extension may also cause pain but is primarily a computer RSI injury, not a climbing injury; however, working this as training rather than prehab may contribute to climbing.
The program focuses on lengthening the muscle under load, which forces the tendon to adapt, while not contracting the muscle under load, which forces the muscle to adapt; the goal is to rebalance the tendon so that the muscle is no longer too strong for the tendon.
These exercises can be done twice a day, with 2 sets of 20 reps each.
It's worth noting that strengthening these muscles is a worthwhile pursuit as well, but with the goal of building grip strength/stability rather than preventing or rehabbing tendinosis.
A final note: in rehabbing my shoulder and then getting back into adding weight to pull-ups, I ran into an imbalance which the dodgy elbows program was unable to correct, which was causing pain in the inside of my elbow. After a lot of trial and error, I finally was able to correct this with skullcrushers, which indicates that this was probably caused by weak triceps.