A friend of mine had a name issue. She wasn't married (really wanted to be, but DP whilst committed was anti marriage) but didn't want to have a different name to her children.
She decided that the children would have her name. She wasn't prepared to have a different name to her own children. Her DP wanted them to have his name (as did his family) but she said that was only going to happen if she had his name too. Although she never said it, I think she took the view that her man would only have the right to name his child if he married her. She never demanded he marry her or anything like that, but made it clear that the 2 things went together. Tbh, I could see her point - why should the man get to name his child, if he isn't prepared to marry the mother (assuming the mother actually wants to get married)
So friend and children had one name and father had another. She felt happier doing the school run etc having the same name as her kids.
After about 10 years, DP decided he would marry her, because he knew it was important to her (turned out he had a fear of the marriage ceremony not marriage itself). At that point they all took on DPs name.
I think a lot of this name thing for many people is related to whether they want to he married to the father. It is certainly more complicated if there have been previous partners or husbands who were fathers. My personal view of the rule of thumb is that the mother should decide. Generally speaking, I don't think the children of previous relationships should have their surnames changed unless there is no contact with father. Both mother and children of new relationship should be able to choose which names to have - children very likely to have name of father if married, (might or might not if unmarried - mothers choice) mother might or might not take on mans name for a variety of reasons, such as work name, attitude towards giving up own name etc)
It seems to be assumed by most in this thread that the norm is for the father to give his name to the baby. Personally, I wouldn't make this assumption unless married. (And even then recognise it is mothers choice) If parents aren't married I can see no reason why the baby should automatically have his name - it is totally the mothers choice then. I see these names as uniting families together - ideally a united family will all have the same name - for many people they don't need to be married to feel fully united, but for others they do and of course a woman only takes the mans name when married, so understandably may not like baby to take mans name when she herself is left out of that naming unity.