Ataxx & Hexxagon
Ataxx is an abstract board-game-as-video-game that borrows heavily from Reversi while adding just enough changes to make late game reversals more strategic. Hexxagon is a variant played on a board with hexagonal squares; it's the first version I saw and one of a large number of clones and spinoffs. The Wikipedia page has an overview of most of the published ones.
Hexxagon looked like this:
The flyer for the arcade version of Ataxx was amusingly gruesome for an abstract strategy game:
The original game was developed in 1988, prototyped on a chess board with paper pieces, and had a complicated path to release. An extensive but now-unmaintained fan page has a detailed post explaining the origins of the game. Ψ