Is there any other meaning for the name 'Harrison' other than son of harry?

Here's what I found on a quick hunt of the internet.

Most sites say just "Son of Harry"

One site said "Fortune hunter"

Here's the thing about the "son of Harry" thing, though. Harry is a form of "Henry". Henry means "Home King". Therefore, Harrison, through all those links, COULD mean "Home Prince", right?

Thank you