Back in 2007, an USA Today article reported that more men were now taking their wives' last names upon marriage. Two and half years later, the same newspaper now reports that about 70% of Americans believe that it is beneficial for a woman to take her husband's last name when they marry, and, more surprisingly, about 50% of Americans think that the government should require women to do so. From an evolutionary biological perspective, it doesn't matter whether the woman takes the man's last name, the man takes the woman's last name, or the man and the woman each keep their last name after marriage, as long as the children take the father's last name.
Paternity uncertainty is a significant adaptive problem for mammalian males who make extensive parental investment in their offspring (as men do). How can men ever be sure that the children that their wives have given birth to are genetically theirs? Men who are cuckolded, and invest in the genetic offspring of another man, lose both the resources they invest in the children and an opportunity to pass on their genes to their own genetic offspring.
As I explain in a previous post, nature may or may not help the father convince himself of his paternity by making the baby (kind of) resemble the father rather than the mother. However, as I explain in another post, people (especially maternal kin) appear to help, by telling the father that the baby resembles him, regardless of whether it does or not. (Ask any parents of adopted babies, and they will likely relate anecdotes where strangers approached them and told them that the baby "looked just like the father.") After all, the maternal kin, unlike the paternal kin, have no interest in finding out the truth. They know that the baby is genetically related to the mother for sure - there is no such thing as maternity uncertainty - and all they want is to make sure that the father is convinced of his paternity enough to invest in the offspring, regardless of whether or not he is the actual genetic father.
The convention of giving the child the father's last name is another means for the mother and her kin to convince the father of his paternity. Even in societies where the married women routinely keep their last names when they get married, or even among many western professional couples these days where the woman keeps her name after marriage, the children are almost always given their father's, not their mother's, last name. (This is most conspicuous in Iceland, where everybody's last name simply means the father's son or daughter.) By giving their children the father's last name, these women are unconsciously saying, "Honey, it's yours" (even, or especially, when it is not). They need to reassure their husbands of their paternity, but do not themselves need to be reassured of their maternity.
If men whose children have the same last name as they do are more likely to be convinced of their paternity, then, all else equal, they should be more motivated to invest in their children, and, as a result, the children on average should do better than those who do not inherit their last names from their fathers. Children who do not inherit their last names from their fathers are expect to do less well, for example, less likely to survive and reach sexual maturity, than those who do, which is precisely why most children do inherit their father's last name.
As I mention in the earlier post, Russians have perfected this system by going one step further. They give their babies - both boys and girls - both their middle and last names after the father. The widespread practice of patronyms in Russia suggests that Russian men have historically had greater needs to be convinced of their paternity than men elsewhere (all of whom suffer from a degree of paternity uncertainty to begin with). Why is this? There are at least three (mutually nonexclusive) reasons for Russian men's greater needs to be convinced of their paternity. It could be: 1) Russian men's paternal investment was particularly more valuable, possibly because of Russia's hostile environment (Note that both Iceland and Russia are in very cold climate); 2) Russian men, for some reason, have had inherently lower motivation to provide paternal investment in their putative children; and/or (potentially precipitated by the fact that) 3) Russian women have historically been more likely to cuckold their husbands, by being more likely to have extrapair copulations and pass on their resultant offspring as their husbands'.
Now, this is personal....