Hi, Linda,
You wouldn't believe the number of golfers, tennis players, and strength trainers I see and treat with this problem. It ususally stems from overuse and/or inadequate stretching and strengthening of the wrist and long finger extensors (lateral elbow tendinitis/epicondylitis) or the wrist and long finger flexors (medial elbow tendinitis/epicondylitis).
Rest and ice are a good idea. Compression and elevation are good also--doesn't have to be a sprain to use the C,E components--if there's inflammation and pain, there most likely is swelling which the compression and elevation work to combat, too.
Make sure you pay very close attention to keeping the wrists neutral with every exercise you do (except wrist/forearm strengthening, of course), including transporting weights from the rack to your exercise area and carrying objects in your daily life. Stretching this area well will be vital to recovery also. And, there are a few self-massage techniques that are helpful.
If you would like the stretching and massage techniques, please e-mail me. In the meantime, RICE, anti-inflammatory meds if you can take them, and you might want to consider one of the devices worn just below the elbow designed to help with tendonitis. Neither this device nor the anti-inflams are permanent solutions, but they will help with the symptoms until the stretching, strengthening and massage techniques fix the problem.
Take care!
Maribeth