Hellobee Boards

My best friend gave me this one and I love it: http://www.amazon.com/Wherever-You-Are-Love-Will/dp/1250017971

Llama Llama Red Pajama was one of my faves that I got as a shower gift

A couple of LO's faves that aren't the classics are:

My lucky little dragon
Little blue truck
Pout pout fish
Pouch (we got this thru the Dolly Parton library though so not sure if they would get it that way free)

Any Sandra Boynton book! Some of our other favorites are Little Blue Truck, Giraffes Can't Dance, On The Night You Were Born, The Very Hungry Caterpillar

Lots of great suggestions already. Sandra Boynton is great. I'd also suggest her "Going to Bed Book". That's a staple at our house, we've read it every night to DD. Other favorites in our library are (all in board book version):
The Very Hungry Caterpillar
Goodnight Moon
The Runaway Bunny
Brown Bear, Brown Bear
Sometimes I like to Curl up In a Ball (love this one!)

I usually give Mem Fox. We like Ten Little Fingers and Time for Bed. We also like Fiona Watts' That's Not My... series and Karen Katz' Counting Kisses for something interactive.

Fwiw, my sister did this at my baby shower and other than classics, Sandra Boynton was duplicated most.

FWIW, I have a million board hooks, but now DS is past the ripping stage and we don't have that many big kid books. So, I like to gift books for later. I like Library Lion or There's a Bird on Your Head.

Little blue truck is DSs FAVE. He's 5 months and loves it.

On the night you were born is super sweet
http://www.amazon.com/On-Night-You-Were-Born/dp/0312601557

These aren't really classics but DS loves the black and white images and they are supposed to be good for development
http://www.amazon.com/gp/aw/d/1587285940/ref=mp_s_a_1_8?qid=1459818042&sr=8-8&pi=AC_SX236_SY340_QL65&keywords=black+and+white+board+book&dpPl=1&dpID=51WoDFU0ZtL&ref=plSrch

http://www.amazon.com/gp/aw/d/1587285959/ref=pd_aw_sim_14_of_5?ie=UTF8&dpID=51JY5aKy4RL&dpSrc=sims&preST=_SL500_SR200%2C198_&refRID=1BR5SRGJM2NFEFZXWXEZ

http://www.amazon.com/gp/aw/d/0688119182/ref=pd_aw_sim_14_of_9?ie=UTF8&dpID=41o76bbCIRL&dpSrc=sims&preST=_SL500_SR200%2C200_&refRID=1BR5SRGJM2NFEFZXWXEZ

The wonderful things you will be is also really cute but doesn't come in a board book
http://www.amazon.com/gp/aw/d/0385376715/ref=mp_s_a_1_2?qid=1459818194&sr=8-2&pi=SY200_QL40&keywords=wonderful+you&dpPl=1&dpID=51q3%2Bu5cenL&ref=plSrch

Toot by Leslie Patricelli! I loved getting that book.

@NCSUchick27: I think they are the cutest! I also like Chicka Chicka Boom Boom and Brown Bear Brown Bear.

I'll See You in the Morning is one of my favorite bedtime books to read to my DDs. I've given it as a gift many times.
http://www.amazon.com/Ill-See-Morning-Mike-Jolley/dp/0811865436

@Banana330: yes! This was going to be my suggestion if the recipient has a good sense of humor! We love anything by Leslie Patricelli.

My son loves all the books from the Baby Touch and Feel line - it has a picture of an item and then they have glittery/shiny/textured bits for them to touch as you go along. Sturdy board book pages too. Especially now at 18 months where he's learning words for things, but the textures fascinated him even before 1 year.

Dear Zoo has been a favorite for a long time. He also likes Blue Hat, Green Hat by Boynton and Peek a Who?