I've done an indie remake of an old arcade game from 1980.
Can I use its name? I thought copyright expired after 25 years but I could be wrong, hence this question.
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You're asking about more than the name, if you re-made the actual game (rhyme unintended). In the US, copyright has been constantly extended, and is now the author's life + 75 years - so no, the game is still copyrighted.
And the name is covered by trademark, not copyright (which also would not have expired - I don't think it does expire). If your name is identical, then yes it's a trademark violation. If it's similar, then it depends - some companies are more litigious than others (See Edge Games, who sued anyone that had a game with the word edge in it, like "Mirror's Edge" by EA.
Activision had to license the name Civilization from Avalon Hill, maker of a board game with the same name.
The only safe way to do this is to buy or license the rights for the game from the current owner. This can be expensive and time-consuming. The rights holder might ask for a fixed fee plus royalties, and as an indie game developer, that's likely going to be more than you expect to earn.
On the other hand, if the original game was produced by a very small company or an individual, you can probably contact them and get permission to do the remake pretty easily.
If you do a remake of a classic game, the rights holder can sue you and has a reasonable chance of winning. Tetris Holding is renowned for this.
If you just use the name in an unrelated game whose name is not currently being used for any commercial purpose, the rights holder can still sue you to bankruptcy.
It's slightly safer if you are not selling the game, I think. More likely to avoid notice, and the potential fines might be smaller. But you're only safe if you get an agreement with the rights holder.