I wouldn't go with the doll idea. My sister-in-law bought an expensive Barbie for each of her girls for every occasion, but they had to be kept on display and often in their packages, so the girls got frustrated not being able to play with them. Eventually I think they were sold. I have a large doll collection that my grandmother started as an adult, but it's from her travels (and those of other family members) around the world - it has sentimental value for that reason, but it's a lot to care for and the display cabinet is quite large. I enjoyed looking at it when I was a child, and still do, but it was a "look, don't touch" thing which I would not have appreciated if she had given these things to me when I was a kid.
The pearls, stock, bonds ideas are good but they aren't things you can give her to "open" - just put them away for her or have the parents put them in the safety deposit box for her to enjoy/appreciate later. The pearls may frustrate her at a young age before she has enough to look like a necklace, and then what if she wants to wear it to school at age 8? Can't have that!
Whatever you give her now, she won't appreciate until she is older. You can start the collection if you don't expect her to open it and appreciate it - I would hate to see you disappointed!
I enjoy a lovely set of silver teaspoons that my grandmother received from her great aunt - matching spoons, each engraved with the year or sometimes her birthday (e.g. 6-3-08) - she started receiving them when she was 7, and had a set of 12 by the time she turned 18. This was a nice start to her married life, but it's nice for any girl's adulthood, especially because single women never have any nice things that are usually acquired as wedding gifts! My mother-in-law had a lovely set of teacups like one of the other posters mentioned - we enjoy using them at holidays and other nice dinners. A girlfriend had a similar collection from her aunt. Again, these aren't things that can be played with or appreciated by a young child, so it might be nice to start her at a later age.
My daughter has a nice picture frame, and the mat is cut with holes and captions for each year of her child's life - every year, you add another picture of the child at that age. In the center is a photo of the child at birth, with the date and birth weight listed. It's fun for the kids to look at how they've changed.
Consider some sort of scrapbook or memory book with stories about YOUR family, your parents and grandparents. You could add a page or two at every birthday, making the stories longer or more complex as she gets older and can enjoy them. Someday, when she is grown and you are no longer around, she will have a wonderful treasury of family stories.