They should have deleted it and just included his name during the Imperial trainee scene. Which I guess goes for a lot of these weird annoying moments… they shelved half the movie but didn’t have any idea what actually worked.
I don’t think they really shelved all that much (just reshot).
Edit: was his father also just called Han?
I think the implication is that Han does have a last name, but didn’t want to keep it because daddy issues.