There are too many of us for names to be a unique identifier. First names are too generic to say anything about us, and they're usually assigned at birth, before any of our traits are known.
Family names are even worse; here's your father's last name glued on to yours, for no apparent reason. Or maybe your mother's. Or, if you want, in Belgium, a hyphenated version of both parents', but excluding any hyphenated part of theirs. This somehow implies that family is important, but only a small part of your family? I don't get that in a modern context.
Iain M Banks had an interesting take on multi-part names in The Culture, with the naming scheme at least featuring a chosen name. But sadly also including things you have no control over, like place of birth and parentage.
Letting everyone pick their own name would be nice. That's apparently already possible in some parts of the world, but not everywhere. And often you can only change it if your name is truly ridiculous or if you share a name with a serial killer or some such.