When they first opened the arena, there would be an annual UNH v. Dartmouth game hosted there that would normally be dubbed the Battle for the Riverstone Trophy. I think that lasted for a few years, and then the "Riverstone" label simply migrated to whomever UNH would play in their annual Manchester game. I'm guessing that if UNH plays Dartmouth, then it's for the "Riverstone Trophy", while when they play someone else (UMaine) instead, then it becomes the "Riverstone Classic".
As discussed in the thread for this week's lead up to the game, I've always held out hopes this "Riverstone" thing would develop into a NH version of the Beanpot. You old-timers might also recall the short-lived "Governor's Cup" series, which involved UNH, UMaine, UMass-Lowell and (I think?) UVM for a couple of years in the late '90's? I think that was a Thanksgiving weekend tourney, and lasted maybe 2-3 years before interest petered out.
For more info than anyone probably wanted ... "Riverstone" is actually an insurance claims management organization that runs off stranded business lines. It's part of the Fairfax group, and the outfit still operates down the street from the (now) SNHU Arena. Once upon a time, back when I was interviewing for my job with my current company in late '99, I also interviewed with them. I didn't think they had a long term future in the area (which was why I was interviewing in the first place, due to the local departure of my previous employer), but as it turned out, all three of the companies I was looking at are still alive and very active in the area. So I guess I couldn't have messed that one up, no matter how hard I may have tried to.
http://www.riverstone-group.com