Depends on the genre.
In Vampire, set in the modern day and age, all of my characters had first, middle and last names, and, usually, different identities that they had assumed after their mortal deaths. In some cases, for the older ones, several such identities existed, and in other cases, of *really* moldy ones, they only had one name and had been using it for centuries, skipping the whole mortal alias nuisance and having retainers handle the paperwork and such.
In D&D, I rarely come up with surnames, unless there is a specific cultural reason, or a fun nickname that says something about the character. If a character is a member of a noble family, such as the Belabranta of Waterdeep, or House Cannith in Eberron, then yeah, otherwise, I can rarely be bothered. I tend to like names to have a certain 'sound' to them, a certain 'flow,' and often a last name completely bollixes whatever 'flow' was going on with the first name.
A name like Arathilor 'Battlehammer' just comes to a clunking grinding crashing halt for me.
Arathilor Laferanil, whatever. Bleakfist Battlehammer, sure, fine. But mix them up and it's like a musical train-wreck to my ears.