You are absolutely justified in only wanting one removed. Or better yet, you may be able to get just the cyst removed (cystectomy). Surgeons with good cystectomy skills can usually remove cysts and save the ovary or enough of it for hormone production as discussed here - https://web.archive.org/web/20181005231258/http:/ovaryresearch.com/ovarian_cysts.htm.
The large majority of ovarian cysts are benign even after menopause. Post menopausal ovaries have been proven to produce hormones especially androgens for decades (and possibly our whole lives), some of which are converted to estrogen. These hormones are crucial to heart, brain, bone, etc health. The frozen section done while you are under anesthesia should show whether the ovarian cyst is benign.
You did not mention if you still have your uterus. But if you do, it also has lifelong functions so hang onto it. Removal of the uterus (hysterectomy) causes shifting of bladder and bowel increasing the risk of dysfunction (e.g. incontinence and incomplete emptying) especially in the long term. The severing of the ligaments to remove the uterus causes the midsection to collapse causing a shorter and thicker midsection as well as back, hip and leg problems in the long term. Sexual dysfunction is also a common complaint due to severed nerves reducing sensation as well as reduced or lost orgasms / loss of uterine contractions.
The large majority of hysterectomies and oophorectomies are unnecessary. And most surgeons fail to tell their patients of the many increased health risks of organ removal. Surgical consent forms are usually fairly open-ended giving the surgeon license to remove whatever he/she desires versus what the patient wants.
My gynecologist used ovarian cancer scare tactics to get me to consent to hysterectomy and bilateral oophorectomy. I had a history of 20 years of trust in him and wish I had listened to my inner voice that told me something was not right. I also wish I had revised the surgical consent form stating what could and could not be removed under what circumstances (based on frozen section results) and made sure he signed off on the revisions.
I would be happy to share more if you'd like.