Last Names Are Obsolete
In US and similar cultures, last names are family names. Traditionally one shares one's last name with the family they live with, whether they're a child or a parent in that family. Last names are handed down to one's descendants (hence the traditional exception to the wife-changes-her-name rule when doing so would cause her maiden name to go extinct). Your last name functions to designate you as a member of a certain genealogical grouping -- I'm Stentor, of the Danielson family. The custom of a wife changing her last name was really a custom of a woman leaving her birth family and joining her husband's family.
But that's hardly the only way that a surname can be established. Over the summer I did some amateur genealogy. The English segments of my ancestry followed the last-name-as-family-name pattern, such that you could safely predict that each person's father would have the same last name as them, maintaining the names generation after generation. But as soon as any line crossed back to Sweden (mostly in the late 1800s), last names suddenly became true patronymics. Sven Persson would name his son Arvid Svensson, not Arvid Persson -- and his daughter would be Linnea Svensdotter. (At one point I ran across a long string of alternating generations of Bengt Martensson and Marten Bengtsson.) Here, last names indicate your position within a family tree, but don't designate you as part of a specific family group. This makes sense given that the Swedes in question were largely peasants, and so there was no great family patrimony or inheritance to be handed down within the family. These patronymics were then frozen into family names at Ellis Island, such that my last name is literally "Daniel's son" even though my father is named Rodger.
Looking at the origins of other common family names reveals additional ways of establishing last names. You could look to the person's occupation (Smith), or to some notable characteristic they have (Schwarz, meaning "dark"), or to their hometown (Wiener, meaning "from Vienna").
In the modern US, there is no longer much need to designate someone's membership in a family lineage. It's a convenience to have the same last name as one's immediate nuclear family (albeit a convenience it's easy to do without). But when we go beyond that, our notions of relatedness are resolutely bilateral, not patrilineal -- I consider myself equally related to my counsins who are Eberls, Pearsons, Johnsons, and Derrs as I am to those who are also Danielsons. Moreover, nothing of legal importance turns on sharing a last name, or sharing the patrilineage that the name (if inherited by traditional rules) implies. So the notion that last names must be in some way inherited is senseless in our cultural context.
It's unlikely we'll do away with the idea that people should have two names, since it's so deeply embedded in our bureaucracy and computer systems. But there's no reason that we have to stick to the family-name model when confronting the perennial question "but what will you name the kids?" If my wife and I were to have kids, there's no reason we shouldn't name them Billy Stentorsson, Susie Browneyes, and Alex Pittsburgher.