There are tons of interesting mechanics you can integrate into a static board.
For a positional-based game like Chess or Checkers, you could add terrain or interesting shapes. Note for example how much more interesting the Stratego board is because it has those two lakes in the middle, creating three major choke points.
For a territory-control game like Risk, structuring the territories to form interesting shapes or clusters and unique landmarks on the board is more interesting than something purely symmetric. In Risk, both South America and Australia are easy to defend but give only a minor bonus, making them natural areas of early-game contention. In Pandemic, South America is an interesting region because it has the only dead-end location that's hard to get to, while Eurasia is also interesting because it has a lot of interconnected cities that lend themselves to major outbreak problems.
For a roll-and-move game like Monopoly or Life, make the spaces you land on interesting by having them give the players meaningful choices. This is one of the major reasons why the board game Talisman is marginally more palatable than Chutes & Ladders.
You can also, of course, include mechanics that change the nature of the board over time. Maybe you have some tokens that get placed on the board that make certain spaces temporarily (or permanently) impassable, or adding some additional effect to visiting those spaces. That keeps the board dynamic, even if the original starting game state is always the same.