"Gem Theater" is just a name for the theater. Like "Royal Theater", "Grand Theater" or "The Apollo" etc

It depends. It may have some special significance. A gem is a precious or semi-precious stone, so something valuable and precious. So maybe the theatre is an important place. Gem can also be used to signify an outstanding person or thing, which again would be a nice reference to a theatre, to suggest good quality performances!

There are fashions and trends in naming things like theatres, companies, etc. "Gem" is just a way of suggesting something good, expensive, high-class, polished, sparkling, etc. What I find interesting is that "Bijou" was once a VERY popular and common name for theatres, and it also means "gem." It was so popular that in novels from the forties people say things like "the local Bijou" meaning the local movie theatre. Maybe there was a time when theatres liked the "jewelry" association. "Gem" is also an out-of-fashion name for stores of all kinds. One can easily imagine any little store calling itself the "Gem Variety Store" or the "Gem Grocery Store" or the "Gem Appliance Store."