As long as I'm doing defunct downtown restaurants, I might as well do Mayfair Grill as it comes up fairly often. (And often gets confused with the Mayflower restaurant).
Unfortunately, it's another place I know very little about.
I can say that it first listed in the November 1954 Southern Bell telephone directory, and last listed in the January 1984 edition. They never ran a full yellow pages ad, but did have a slogan under their listing for a number of years, one which I seem to have misplaced for the nonce.
I can also say after consulting an old city directory that the property at 343 South Main was another restaurant, called Louis' Grill immediately before Mayfair Grill opened.
Here's what commenter Peter Hoffman had to say about the place:
The “Mayflower Diner” was on South Main Street, across the tracks from the station, on land that’s now part of the Swearingen Engineering Center property (the empty space between the phone lines and the track).
It was a great place to eat. A complete cross-section of Columbia would be there from students, to blue-collar, to state politicians. The food was really good ‘meat and three veg’ and served on willow-pattern plates and the prices were something a student could handle.
It’s been about 30 years since the place was demolished in the name of ‘progress’ but I still miss it.
And here's an amplification and slight correction from commenter badger:
Oh, and that little diner on South Main was actually the “Mayfair” Grill. I think it was open till around ’85. Across from it was the old SCE&G headquarters, currently occupied by USC.