I've never heard of giving the new dad a gift but I have heard of a "push gift" that's for us though n all our hard work during labor...if he does bring up the topic or asks me for suggestions on my push gift im hinting toward a juicy cotour diaper bag (that's a luxury I can't bring myself to purchase for myself ) ;o)
My fiance mentioned gifts the other day, it was weird because I have never heard of anyone giving gifts on that day, but he said a lot of people do it. He said he was going to bring my flowers and whatever candy I wanted and then asked what I was getting him. I never gave it any thought before that moment, but I told him "I'm giving you a baby", then I smiled and he smiled :)
I've never heard of giving gifts on the birth day but for my husband's first father's day, I got him a wooden cube with an engraved saying about daddies and their little girls that holds 2 wallet-sized photos. Each year, I dig out 4 cute daddy/daughter photos to update the cube. I am fortunate to have a very sentimental daddy in my life so I'm not sure how appropriate it would be for others.