E.L.
It depends on how active your 1 year old is, but a friend of mine did hers at Chucke Cheese. (This is in IGH) There is a toddler area with a little slide and some rides, a video game area, and a big climbing thing for kids in between. I've been to a couple of birthday parties there now and it seems easy and is always fun for the kids. I think it's pretty inexpensive too.