We had the same question when our daughter turned one five months ago. Boy the time goes fast! We got her a ride on toy-actually two, one was the gobble 'n go hippo by fisher price, which she can ride or walk behind, it was great when she was learning to walk and she still loves it now. Her great-grandparents got her the new Radio Flyer ride on toy without pedals, but it has real steering and a bell. She still isn't that coordinated with it and still can't make it "go". She just gets on and off and plays with the bell. I think I would recommend that for Christmas for your daughter. She got a wooden rocking horse for Christmas (at 15 months) and she really loves that now. I've given bath toys and books for gifts in the past and they have been big hits. I'm not a person who wants a million toys in my house, but there are a few things that kids of this age need to help them along developmentally- a ride on, something to walk behind, something to pull behind them, and books with big pictures. The caterpillar with the balls by FP is a great pull toy, and the traditional popper is a big hit for little kids. My daughter walked around with the popper off the floor for a couple of months until she got the hang of it, but she still had fun. Now she has a blast chasing the dog around with it! We got our daughter an outside swing that she absolutely loves. A climber (like a Step 2 Kangaroo Climber, or one of the Slide and Climbs) is a great toy as well, especially one that you can use indoors since we get a lot of rain here. I just ordered one for her today, but I wish I had bought one for her a long time ago, because now that she can walk she is climbing on everything in sight! Lego Quattros are nice big blocks, and the Melissa and Doug wood puzzles with 3 pieces are also a big hit. She loves peek-a-blocks, books, the little people sets, and her toddler music as well. Some of the things I am listing are a little beyond a one year old's capability, but she will grow into it quickly, and will be able to use all of this before Christmas rolls around.
My parents gave her a plaque with her name and its meaning on it, and that is something she is going to have for a very long time. A few other keepsake/non-toy ideas for you- a music box, a Precious Moment-or you could start the Precious Moments birthday collection, a first Bible, or start any other kind of tradition you would like. My cousin always got a pearl every year to add to her necklace (although I think she started this when she was about 8, I don't remember). If all else fails, savings bonds or money in a savings account will always be appreciated by her later. Have a great party!
I just went back and read the other posts- my daughter loves the real swiffer, too, and I've shortened it so it is just her size. Starting her on housework early! =) We also did get a really nice wood toybox from her other grandparents, and keep that in the living room. That has been a big help on keeping the clutter down.