I had not one, but TWO birthdays (10 and 11) on the Battleship at the
U.S.S. Massachusetts / Battleship Cove in Fall River. SO cool. you get a private room in the galley area (they serve lunch and Cake- provided by you). You get to tour all three ships and play around on them. It's adjacent to a nice waterfront park which is fun for feeding ducks and such too. They also have a nice Carousel overlooking the bridge and Mt. Hope Bay.
As has already been suggested, any zoo is cool for this too (I'm partial to Roger Williams in Providence).
The Brockton
Rox (Single A Baseball) are AWESOME for parties. They do wonderfully with group events. I've always been impressed with them
Dave and Busters (located in Providence Place mall) may be a BIT too much, but it's not bad for next year maybe. The Charlie Horse is good and along these same lines.
New England
Aquarium may have some deals for Birthdays and so might the Museum of
Science. I have no experience with them, but can only imagine that if they do host events, they're a lot of fun.
Edaville
Railroad isn't a bad idea, but they're not open until June 20 so it may not work if the birthday is earlier (I'm assuming it is?).
Out of all of those, I LOVED the battleship. It's so cool. Fall River may be a bit of a hike, but the battleship is a great experience at that age (you can literally explore the entirety of the interior and exterior). Any Zoo or Aquarium may be fun too.