Because there are so many great books available, the most important thing is to determine the favorite activities, subjects, and interests of your niece and choose a book reflecting these. She is more likely to want to read whatever book you get her if she's enjoys the topic.
Once you have determined what she likes, you may want to take her on a trip to your local library and get a better idea of what types of books interest her. Once you've determined that, you may wish to go to the bookstore alone and spend a moment browsing through the children's section.
At 3, your niece will probably be more interested in colorful, interesting pictures that have a simple storyline. Whatever book you pick may very well become a favorite of hers, if you take the time to find one that is about something she loves.
I work in a nursery school and some favourites of the children are The Very Hungry Caterpillar, The Gruffalo, The Gruffalo's Child, Winnie The Witch, Room On The Broom, The Cat In The Hat, Green Eggs and Ham, The Gingerbread Man, Goldilocks and The 3 bears, Aliens love underpants, The 3 Little Pigs, Peace At Last, 5 Minutes Peace, Handa' Suprise, Whatever Next!, Dear Zoo, Spot the Dog, Duck In The Truck and Threadbear!, have fun shopping for books and I hope the bookshop has plenty of choice.