When it comes to naming a character, the first name is most likely going to be the most important thing for you to figure out. But if you look at your characters as if they were real people, then their first names really only make up part of what their full names should be. And if you want your character to feel like a real person, then he should have a last name.
Of course, last names usually won't be as important as figuring out the first name. It won't be used as often. Depending on the character, it may not be used at all. Every single character who pops up in your book doesn't necessarily need a last name. It would probably be distracting trying to keep track of all of them. Your minor characters probably don't need them, but it's probably a good idea to give last names to your main characters. It may come up, and it may not, but it's good to have it in the back of your mind.
You probably won't be as passionate about a character's last name as you will be about their first, and that's perfectly fine. It doesn't always make up a huge part of their identity like a first name does. It can be something common or generic and still not make the character seem boring. What it really does is make them feel like a real person. Even if you don't use the last name, it's a good idea to know it so they feel more real, more concrete in your mind. And if you need to sneak it in somewhere, then you already have the name on hand and don't have to spend a lot of time searching for one.
FUN TIP: Can't come up with a last name? If you've got a phone book lying around (yes, they still exist...), close your eyes, pick a random page, and point your finger somewhere on the page. If you like the last name you land on, use it!
So when will you actually use a last name within a story? Some people go by their last names more often than their first. At my last job, pretty much everyone called me Foster. So whether your character goes by his first or last name can depend on where he is and who he's with. His family will most likely use his first name (or his full name if he's getting yelled at, but more on that tomorrow...), but maybe some of his friends call him by his last name.
It can also come up when mentioning family members. If your MC's parents are constantly being referred to as Mr. and Mrs., then obviously you need to know your MC's last name. You'll want a last name that actually fits with the first. They shouldn't have similar letters or sounds, or both have a ridiculous amount of syllables. You'll really just want them to flow together, to actually sound like a name that someone would have.
WHAT I'VE DONE: I only gave the two main characters in Uneven Lines last names (plus the MC's mom, of course). One of Jordan's friends, Brian, pretty much always calls him by his last name, Palmer. I don't even know why. He's just one of those guys, I guess.
Do you give your characters last names? Has anyone ever called you by your last name?