An acoustic guitar. "Git" is short for guitar, while the "box" refers to its hollow body.
Usage of the phrase can be found in an episode of "The Twilight Zone" called "The Passersby" from 1961. In the episode, a Civil War veteran refers to his git box, suggesting that this phrase dates back to at least the mid 1800s.
Just sit back and relax while a strum a few on my git box.