There are very few rides a 1 year old can't do. We took DS at 5 months and the only ride we went on that he didn't was test track. That being said DH and I aren't big roller coaster people so it may have been a bit easier.
Here is a list of the rides that have a height requirement:
Magic Kingdom
Space Mountain 44 inches
Splash Mountain 40 inches
Big Thunder Mountain 40 inches
Stitch 40 inches
Barnstormer 35 inches
Epcot
Soarin' 40 inches
Mission Space 44 inches
Test Track 40 inches
Hollywood Studios
Rock'n'Roller Coaster 48 inches
Star Tours 40 inches
Tower of Terror 40 inches
Animal Kingdom
Dinosaur 40 inches
Expedition Everest 44 inches
Kali River Rapids 38 inches
Primeval Whirl 48 inches
As you can see it means that there aren't a lot of rides in Disney Studios, but other then that there are a ton of rides you can do. We had a ton of fun when DS was 5 months old and we went and we're heading back next fall with a 6 month old again and DS at 2 and a half.
Oh and tickets are free until they are 3. Which is really nice and a great way to save some money by taking them before that point.