Ezreal Wright (sorry bad reference to an old Boards user -- there was also an Ezreal Edgeworth lol)
To be serious, while it would be interesting to find out and we can speculate using the name of his uncle, that's not necessarily accurate (most of my uncles don't share a last name with me for example because they are my mother's brothers - could be the same with Ez). In reality, we know a lot of "last names" for various champions, but in the grand scheme of things looking at the majority of champions, we don't. Just using Piltover/Zaun champions as example, we don't know the last names (to my knowledge) of Jayce, Caitlyn, Vi, Jinx, Ekko, etc... (although we could use Camille to speculate Caitlyn's name possibly). That's probably because last names aren't necessarily important in the development of a character. We get more names in places like Demacia and Noxus where there are some power families, but you don't have that as much in places like Piltover. It's even less likely that his family was an influential family given his whole motive to his actions is to get his name famous enough that his parents find him/come back for him. Wouldn't really be necessary if his name was part of some high class well known family.
If he was the only champion from Piltover that lacked a last name, I'd be more suspicious, but since he isn't, it's probably not as important to Riot (yet).