Upvote:4
Actually, it turns out that the regular hostel search engines do allow you to search for private rooms specifically, and sometimes for family rooms even more specifically. But on all of them you have to do an initial search, and only then can you access the "filters" that let you specify room type. This was not straightforward to me, and I thought I was pretty experienced at hostels and Internet. Maybe this is completely obvious to some people but hopefully this answer will still be useful for other people like me who didn't realize you could do this.
HostelWorld
Does let you look specifically for family rooms.
HostelBookers
Doesn't let you look specifically for family rooms, but you can at least search for private rooms for 4 or more people. After that you'd have to dig further into the page for each property.
Hostels.com
Does let you look specifically for family rooms.