My little one loves to play in the water so we got her a water table for her 1st birthday and it turned out to be a great present. When its a nice day, I fill it up for my girls (I have a one year old and four year old) and they just go to town splashing and what not.
We got some great practical stuff for her birthday from friends. One of my friends gave her teething tablets and eczema cream. We also got sand toys for playing outside and some clothes.
My oldest when she turned one, got a little sofa the one that folds out into a bed and she just LOVED it.
It is really hard buying for a first birthday because they arent old enough yet for the really fun stuff but all the things for their age range are noisy. I cant stand any more noisy toys. Thats just me though.
I really like the suggestion of sippy cups and other "hardware". My 1 year old really likes the nuby straw cups. They are fabulous and the only things she will drink out of that dont spill easily. Maybe even a new bib. I really like the bumkin ones that are wipeable and have snaps on the crumb catcher that you can easily clean. They also have knock off ones at Walmart that are much cheaper.
Walking shoes would be a good present too. Get a good brand like stride rite or the new walker shoes made by robeez. Anything with a soft sole is what the pediatricians recommend for walking now, not those hard shoes that we wore as kids. My daughter got a pair of stride rite shoes too. So cute, I got them at Ross on clearance for $9.00. I hate spending alot of money on kids shoes you know. But I like getting good brands.