It is possible to have a cyst on birth control.
Pain on the lower right side may be suggestive of ovarian pain (i.e. cyst, a mass etc.). Other possible causes can include problems with the appendix, irritable bowel disease, or inflammatory bowel disease.
I would consider ovarian causes first, and a transvaginal ultrasound would be the most reasonable test for an initial evaluation.
Followup with your personal physician is essential.
This answer is not intended as and does not substitute for medical advice - the information presented is for patient education only. Please see your personal physician for further evaluation of your individual case.
Kevin, M.D.
http://www.straightfromthedoc.com