One night stand

what does "stand" mean in "one night stand"?

From Merriam-Webster online:

Main Entry: one-night stand
Function: noun
Date: 1880
1 : a performance (as of a play or concert) given (as by a traveling group of actors or musicians) only once in each of a series of localities
2 a : a locality used for one-night stands b : a stopover for a one-night stand
3 : a sexual encounter limited to a single occasion; also : a partner in such an encounter

could stand be replaced with other words?

Yes, try "occurence".