A.,
First the question is how big is it going to be. Who are the primary people coming - adults or kids. Really the first birthday is more for the parents and grandparents than for the child so try and keep that in mind. I would also suggest scheduling around her nap. Some kids at 1 year are still on 2 naps and some are on 1 nap so I would wait til closer to the event before selecting a time. You don't need a meltdown because it's nap time when you are trying to sing happy birthday. I don't know where you live so I will tell you a few places I know of.
As for places -
Together in Motion in Arlington has a nice big open room for kids to play in, romp around and crawl.
Chucky Cheese in Arlington or Cambridge - I forget which.
Or just a nice restaurant to have a lunch. Some restaurants you can rent a room and have a private party in - which might be your easiest option.
Good luck,
L. M