What is the value of a 1788 New Hampshire quarter?

Unless it's uncirculated, face value only.

To explain, your quarter wasn't minted in 1788 nor was it made in New Hampshire. It's one of the State Quarters series minted from 1999 to 2009 honoring each of the 50 states and 6 other districts of the US. 1788 is the date New Hampshire was admitted to the Union; the minting year (2000) is at the bottom of the reverse side.

Hundreds of millions of state quarters were minted; except for a few issues that had minting errors any that you find in change are only worth, well, a quarter.