HoMM3 HD is a mod (add-on) for the game Heroes of Might and Magic 3.

Then, a simple, universal option would be [map name].112.gm2, as it would also be suitable for offline single-player.

What's wrong with my suggestion of adding the generation number? I think saving based on that number is convenient, as it preserves saves from restarts.
+I think it would be useful for tournaments to have the option to reset the current player's timer to a specific value (or by a number of seconds/minutes:seconds) when loading or during the game. Or an option to add the already played time when loading... Don't you think it's a bit strange to skip this when using a mod aimed at saving time and convenience?