Realistically, the Mayo name may or may not open doors, once you graduate. Folks "in the know" realize that the mothership in Rochester (MCR) has a different culture from MCS and MCJ. Neither Scottsdale nor Jax seemed to be as "rigorous" in expectation as MCR, at least not when I was there. MCR placed much more emphasis on ITE and board scores. They also seemed to think much more highly of themselves.
I was part of the 2nd class to graduate MCJ and feel that my training was superb. Most of the attendings at Jax and Scottsdale came from Rochester. I had 3 oral board examiners to quiz me every single day of residency (Cuchiarra, Murray, Brull), as well as Dorsch and Greengrass. There are (were) many other well-published and competent attendings.
At the end of the day, depending upon your goals, it matters little. Outside of the world of academics, the name of your residency institution will not matter all that much. Your individual skill set, work ethic, and ability to play well with others will speak volumes. 🙂 Those are the things I look for when we interview candidates and assess potential new hires.
Best of luck!
PMMD