Kathy, I think books can make lovely presents for babies, although they might not appreciate them until they are getting on for a year old(?)
The books my children enjoyed when small (or maybe I enjoyed reading to them) were The Very Hungry Caterpillar by Eric Carle, Dear Zoo by Rod Campbell (with lift-up flaps) The Baby Catalogue byAllan and Janet Ahlberg and the 'Busy Town' series by Richard Scarry.
All good as there are very few words, which children can soon recite with you, interesting pictures which they can point to and, I think, most of them come in both 'board' and 'paper' editions. Board good for chewing on and paper for tearing!
I never much cared for the 'cloth' books, too floppy and limited in words/pictures.

Goodnight Moon, by Margaret Wise Brown. Available in board and hardback. A real classic.

My daughter loved the Richard Scarry books when she was a baby. I also bought a very nice copy of nursery rhymes -- she liked those, too. I frequently gave them as baby gifts.

I'd suggest the Sandra Boynton board books, my kids loved The Going to Bed Book, Moo, Baa, La La La, Barnyard Dance, Oh My, Oh My, Oh Dinosaurs!, Doggies, etc. They also loved the Eric Carle books, especially Brown Bear, Brown Bear What Do You See? and The Very Hungry Caterpillar. I still give these as baby gifts, since everyone seems to receive dozens of tiny socks and onesies but very few books :)

I bought each of my great-grandchildren Beatrix Potter boxed collections and the CDs to listen to until they could read them. You can get the character pictures for the nursery too. Also lots of other items. It is a very thriving industry and can keep you collecting crockery and toys for many birthdays etc.! I once found a beautiful stuffed squirrel with a thick bushy tail which made a good Nutkin.

My nieces/nephews loved The Velveteen Rabbit by Margery Williams, and Old Possum's Book of Practical Cats by T.S. Eliot. A couple of others they loved when they were a bit older (and now read to their kids) are The Good Master and its sequel The Singing Tree by Kate Seredy. Her illustrations are lovely, too.

For toddlers, the Don and Audrey Wood book, Piggies, is adorable -- great illustrations.

What a lot of good ideas you all have. Thank you! And Rosefolly - good thinking about keeping track of what I've already bought for a child. I would never have thought of that.

I also want to thank Rosefolly for that wonderful idea!

A big hit with some of the little ones in my family was Is Your Mama a Llama by Deborah Guarino. One would ask his mother to read the Mama Mama book. The Beatrix Potter books are good for children a little older, as is a book of the tell-me-a-story kind, i.e., The Three Pigs, Goldilocks and The Three Bears, Little Red Riding Hood, Billy Goat Gruff, etc.

I second the Sandra Boynton books as well as Good Night Moon and a touch & feel Pat the Bunny. I'd add one of the Rosemary Wells/Iona Opie books of nursery rhymes--the illustrations are charming-- and a book of fairy tales for the youngest listeners.

Maina, I loved those two Kate Seredy books, too.

I read Christina Rosetti poems to my kids, there are some delightful children's poems. We also did a lot of storytelling poems. And ditto the Sandra Boynton books, so rollicking! Learning about rhyme is a good basis for starting to read.

Both my grandchildren loved Baby Talk, which is a lift the flap book. They love the pictures of babies and love learning to lift the flaps. The three-year-old still likes this book.

http://smile.amazon.com/gp/product/0756609860/ref=oh_aui_search_detailpage?ie=UTF8&psc=1

Another is Colors (Bright Baby) which is a board book as well.

PS Just thought of another. There are plastic baby photo albums that you can put family pictures in for babies or young children to look at the photos, talk about who it is, etc. This has been popular with my grands too.

Here's one: http://smile.amazon.com/Babys-First-Photo-Family-Friends/dp/B000Q7G3VM/ref=sr_1_1?ie=UTF8&qid=1432079810&sr=8-1&keywords=baby+photo+album