standalone pump options for leaky below-grade room
First off, your friend should definitely talk to his landlord about the flooding. Even if the landlord won't address it, it's possible your friend could be held liable for water damage from the flooding if he doesn't notify the landlord.

There are two big differences between the pump you need here and a normal sump pump. The first is that it won't be run submerged; most sump pumps (and many other kinds of pumps) need to be run submerged to prevent overheating. Second is that it needs to be able to suck up very shallow water, which generally means it needs to be self priming as well as needing a specially designed inlet and float valve.

Pumps designed for that kind of task are typically called a "puddle pump".