According to Solo: Tales from Vandor (a canon book containing various legends that are told about Han Solo and the protagonists of the Solo film) the Imperial 'intake forms' require a surname without exception. Solo is a common workaround for individuals who have no second name or are unwilling to share it.
Then just the other night, this retired Imperial intake officer said the Empire's standard Military intake form requires a last name, and the computer system rejects any application if the last name is left blank. So if you come from a culture where people only have one name, Joining the Empire means getting a second one whether you like it or not. She said different intake officers -fill in different last names, and two of the most common choices are NA - for 'not applicable' - and SOLO.
If that’s what happened to Han, I guess he's lucky. He could have become known as Han Na
Han seems to consider his name to be a mononym. His entire family are dead and he doesn't feel sufficiently connected to anyone else (aside from Qi'ra) to use their family name as a replacement.
The man waited, but Han didn’t say anything else. “Han what?” he asked.
Han frowned, confused. Han was his name. It had always been his name. He didn’t have another.
“Who are your people?” the man pressed.
His people? His family was gone. The White Worms hadn’t been family. The closest thing he had was Qi’ra and he didn’t know her people’s name, either. Neither seemed to be the answer the officer was looking for.
He shrugged. “I have no people. I’m alone.” The words hurt more than he had expected.
Solo: A Star Wars Story: Expanded Edition
In the film's junior novel, the implication is more that he doesn't feel that there's anyone who's worthy of him. By refusing a second name, he's cutting his ties with Corellia.
“Han. Han what?” The recruiter gazed at him impatiently, eyebrows raised, fingers poised over the keypad to complete the application form. “Who are your people?”
Glancing over his shoulder, Han cast one last look at the world he was leaving behind. And Qi’ra. That final image of her—her face on the other side of the glass, her voice giving him permission to leave as Rebolt and the others pulled her away from him—would be burned into his brain forever.
Solo: A Star Wars Story: Junior Novel