I'm so sorry for your loss!
You should definately do whatever it takes to put the father's name on the birth certificate. As others mentioned, there may be benefits the baby is entitled to, but only if it is legally noted who the father is. If you and the father are married, by all means, name the baby with the father's (and, presumably your) last name.
If you and the father are not married or engaged, I would think twice about putting the father's name as the baby's last name. I'm guessing from what you wrote that you and the father were not married. While it certainly would honor the father's memory to name the child after him, maybe it would be better consider using that name as a middle name. You need to carefully consider the family you will be. As your baby grows, a natural question is "why is my name different from yours, Mommy?". I know successful business women who had maintainted their maiden names who finally changed after their children wanted to know why they had different names than their parents. It affects their identity of where they belong. You will be the baby's family, unfortunately without the father. It may be better to concentrate on the family you will be, rather than the family you would have been. So, my thought is, if you're not married, then honor the father by using his name somehow, but give the baby your last name.
Our thoughts are with you through this difficult time.