The question is partially answered in the article 25 Random Things About Magic:
Magic was almost not called Magic. In fact, when the first solicitation for the game was sent out the game was called Mana Clash. The reason for this is as follows. Richard (Garfield, of course) called the game Magic when he originally designed it. All through the original playtests the game was called Magic. Then when they went to officially name it, they got a lawyer involved who informed them that Magic was too broad of a word to ever be able to copyright. So they started looking for alternatives. Mana Clash was the name they liked best from that search and thus they used it on the solicitation. But everyone playing the game (Richard, the playtesters and the people at Wizards) still called the game Magic. The name just seemed too perfect. They went back to the lawyer and asked what they could do to call it Magic. The lawyer told them they needed to add something else unique and "ownable" onto it that allowing them to copyright the name as a whole. That is how "The Gathering" was added.
The reason for the specific choice the Gathering is described in this interview with Richard Garfield:
Why is it called Magic : The Gathering?
“Well… there’s a few answers to that. When it first came out, we said ‘the gathering’ was a gathering of friends, a gathering of people, and a gathering of cards. It seemed like a good descriptor for our first set. We were expecting our second set to be ‘Magic : Ice Age’, and to have chapters in this book going along, but that was before we sort of ran them all together. So all the cards are now part of Magic : The Gathering – which originally going to be like the first chapter.
“My name for it was ‘Magic’, and the reason we didn’t use that is simply because ‘magic’ is a hard thing to own, and there were all sorts of really terrible names that were floating around that we could own – but Magic : The Gathering was a good final resting spot because we could own ‘Magic : The Gathering’ and people could still call it ‘Magic’.
An alternative interpretation is that it refers to the gathering of magic/mana, to gather power/force from the world around to cast spells, although I don't have a reference for that.