Unfortunately this is an academic exercise because I'll never make $250,000 profit on a home, but I'd like to know if a minor improvement, like adding a $30 fan in the bathroom ceiling, raises the cost basis of a primary residence.

Thank you,
Houyhnhnm

Cosmetic changes do not increase the cost basis.
Repairs do not increase the cost basis.

If the bathroom had no exhaust fan, and you added an exhaust fan, I would argue that you increased the cost basis by $30 because you added value to the home. If you merely replaced a fan, then that would be a repair and would not qualify.