The connection with all the MGMs in Las Vegas, as I've heard it...

Kirk Kekorian sold the original MGM Grand (now Bally's) and then acquired property for a giant new MGM, complete with theme park, on the grounds of the Marina. It would take a while to build it, and he wanted to recruit and keep his management talent in the meantime.

So, he bought the Marina, Sands and Desert Inn as training grounds for his team in the meantime, naming them MGM Marina, MGM Sands and MGM Desert Inn. When the new MGM Grand was ready to open, he sold each one (except for the Marina, which was incorporated into the new MGM Grand.)

This all happened in late 1980's and early 1990's. Check the Chip Guide for specific dates.