…Each of us kept attacking the same small piece to capture it to our side, and we just kept undoing the previous move. After weeks of analysis, we concluded there was no better move for either of us. Do we just end the game in a stalemate?

A: There is no formal rule barring simply undoing your opponent’s previous move. Sometimes there is some other opportunity to sacrifice a piece somewhere else that could tip things and break the stalemate… but if neither player is willing to change their move, yes, you’ll just have to call it a draw. The good news is that now you get to play another game of Homeworlds!