Hi Innercalm,
This is a big decision and no one can tell you what to do. Bottom line is that you need to trust whoever you go with. My husband just had a minimally invasive mitral valve repair at NYU and he's doing well.
Like Hank said, mitral valve repair is common and there are a lot of great surgeons all over the country. When we were interviewing surgeons we were told by our Cardio to look at NYU or Cleveland Clinic if we wanted "minimally invasive" surgery. Reason being, minimally invasive requires a lot of practice and training, and you want a skilled surgeon who has done A LOT of minimally invasive surgeries. Other hospitals offer minimally invasive but, in my research, NYU and Cleveland Clinic have the most experience.
One other thing worth mentioning is that minimally invasive surgery is still open heart surgery and there is recovery time. Minimally invasive offers some benefits such as less bleeding and a smaller incision, but it's still a BIG surgery. Also, each institution has their own minimally invasive approach. NYU goes through the ribs on the right side of the chest, usually below the right breast. Cleveland Clinic does a mini-sternotomy (they still break the breastbone but it's a smaller incision compared to traditional sternotomy). Some institutions use robotic assistance; others don't. So, it really depends on the hospital.
I hope this helps a little. Good luck with your decision.
Sincerely,