So you're saying you have one cyst and they want to remove both ovaries? Why? You shouldn't need to lose either ovary if you have a surgeon with good cystectomy (cyst removal) skills if the cyst is proven to be benign (as most are).
Ovary removal has been shown to increase risk for a number of health conditions including heart disease, osteoporosis, dementia, Parkinson's to name a few. They produce hormones our whole lives that are essential to good health so you don't want to lose an ovary or ovaries unless it is necessary. Ovarian cancer is actually rare so removing them does more harm than good in women who do not have an increased risk for ovarian cancer.
The uterus also has life-long functions as far as anatomical and sexual integrity as well as hormonal since the uterus, ovaries, and tubes work together.