Literally means "beautiful sojourn", derived from French beau
"beautiful, nice, fine" and French séjour
"sojourn, short stay". As such, this surname is most likely a locational surname, in that it originally referred to a scenic place to sojourn in. Interestingly, there was also once a hamlet called Beauséjour in France, which was destroyed during the First World War. But the hamlet had been founded in 1820 (some sources claim 1825), so it is not possible that the ancestors of some modern bearers of the Beauséjour surname took their surname from the hamlet's name, because all French people already had a fixed and hereditary surname by 1820 (most of them already had one since late medieval times, but Napoléon's introduction of the civil registration in 1792 ensured that the remaining people without a fixed or hereditary surname finally got one).
With that said, it should be noted that there is also a small possibility that (in some cases) the Beauséjour surname is actually derived from a nickname, in that it originally referred to someone affluent who regularly liked to sojourn in a place with beautiful scenery. Perhaps even to someone who was a wanderer or had a profession that required a certain amount of travelling, which would have exposed them to a lot of scenic places on their travels, some of which they would have spent at least one night in.