Do gnrh agonists have side effects?

Absolutely. These drugs can cause a temporary medical menopausal state with all of the associated symptoms, ie, hotflashes, vaginal dryness, sleep disturbances, mood swings, etc.
Few but reversible. The most widely accepted and recognized side effect from prolonged use of gnrh therapy is decreased bone density. This tends to resolve with cessation of therapy. Make sure your child is getting adequate vitamin d and calcium intake to minimize impact.