Hand wrist and elbow problems can be common place for athletes such as climbers. One of the first things I tend to look at with issues such as tendinopathy or bursitis is the function of the scapula-thoracic joint (shoulder-blade and rib-cage) and the gleno-humeral joint (upper arm bone and shoulder-blade). If these are not working optimally then somewhere else normally has to pick up the slack. This can then lead to elbow, wrist and hand problems.
In the video below, Dr Kathy Dooley explains through a simple and effective exercise to work on full range strengthening and mobility of the fingers, wrists, elbows and shoulders. Have a play and even try adding some rotation at the shoulders and wrists.