Yeah my mom hates road bikes for the same reason- she finds them too unsteady.

I assure you, this is not really a problem.
Not to say there aren't more stable bikes out there, but it really isn't a problem. I really don't understand why the general public buys so many mountain bikes-they are so cumbersome to ride around town compared to a road bike. I mean, it's pretty simple: just look at the names. One is made for... Roads. The other... Mountains.