Question:- Does the fathers name have to be on a child's birth certificate?
Reason:- My father never knew the name of his father and I would like to know if there is a way of finding out - he was born in 1916

a fathers name doesn't have to be added to the birth certificate

I doubt it would really be possible, unless they were married.The fathers can disappear into the ether, if their name is not registered, or they could then.

Born during the first world war, could also be a reason for not having had the father's name registered.

What a shame his mother didn't tell him, just once! Are there any other relations who may have inherited a bit of information to throw a little light on the situation? Sometimes an old letter can reveal a lot of stuff. I have two of those, written in 1920's/30's and they have only just found there way to me.
Good luck with your hunting.

If the father is not present at registration the only way his name can appear is if the parents are married.

Going only on your question, have you in fact seen your fathers birth certificate?

If you wish to PM me his name I can do some checks for you.

In answer to your first Question:
1) Where the parents were married to one another, fathers details must be entered in the register and only one parent will sign the register (or some other informant).
2)This situation lasted until the Registration Act of 1875 where the instruction read "The putative father of an illegitimate child cannot be required as father to give information respecting the birth. The name, surname and occupation of the putative father of an illegitimate child must not be entered except at the joint request of the father and mother; in which case both the father and mother must sign the entry as informants".

Let me know if you want me to check his details?

Be prepared - it may not be a 'romantic ending' - this happened to one of my in-laws who tried to find their Grandfather - suffice to say he had wilfully abandoned his pregnant teenage mother and proceeded to do the same to at least 2 other women.