Yes, a half-elf could be half-drow.
Drow and humans can have children, and the resulting children are half-drow. In previous editions, half-drow were a bit mechanically different to normal half-elves, as they had a different set of racial traits based on their drow parentage; in 5e, however, they seem to be much closer. The Sword Coast Adventurer's Guide says of half-elf parentage:
Some half-elves in Faerun have a racial trait in place of the Skill Versatility trait. If your DM allows it, your half-elf character can forgo Skill Versatility and instead take the elf trait Keen Senses or a trait based on your elf parentage:
From this list, you can pick to be a descendent of wood elves, sun elves, of "aquatic heritage", or of drow, with the choice determining which alternative racial trait you might take.
This is an optional choice and thus your half-elf can be of drow heritage without any mechanical differences from the standard half-elf. Strictly, this is specific to the setting of the Forgotten Realms as it's in the SCAG, but there's no compelling reason it would differ in most settings.
You would expect a half-drow to look noticeably different, however, given the black skin/white hair/funky eyes combination that drow have going on - so if this hasn't come up before it might be a bit of a retcon to the character's appearance. NPCs who met a half-drow would usually react quite differently to meeting a normal elf or half-elf as the drow are widely feared and reviled on the surface.