hoping to get a "biggie" locale for one week
Your friends could easily go to Berlin from there for a couple of days, so there you've got your "biggie". What age are your friend's children, BTW?
But I want to make sure we have things to do.
See above. Unless you are set on seeing Disneyland castles and the like (which obviously you aren't), I assure you you won't be bored. My goodness, how silly of me: You've even got your Disneyland castle within easy reach! :-)
If you tell us more about what you and your friends are interested in, we can give you more information.
It's just harder to get info for little random places (to use in the
US at least).
True, American travel guides tend to assume that there is no civilized life north of Frankfurt or Berlin. :-)
That said, I wouldn't exactly call a town of 200,000 a "little random place". ;-)
For some basic (though often sketchy or incomplete) information, I find Wikivoyage a pretty good starting point. For Rostock, they even have a list of links where to "go next" which looks pretty helpful. The Mecklenburg-Vorpommern article also looks like a good starting point to me.
Also trying to keep our costs down.
That home exchange is going to be your best bet on that of course. The Baltic Sea destinations are popular with German families vacationing in summer though, so you will have to live with the fact that it's peak season.