You should put your last name on your mailbox along with your house number to make sure the mail is delivered properly.
While not required, putting your last name on your mailbox can help the mail carrier deliver mail easier and find the proper mailbox.
My mailbox on the street simply has my house number and I've gotten mail just fine.
I live in a small town though and so the mail carrier knows just about everyone here but in larger towns it can be useful to put your last name on the mailbox.
However as long as you have the proper street and house address on your mailbox your mail should be delivered just fine.
The requirements that the USPS has for mailboxes includes.
Address number: All curbside mailboxes must include the address number.
Street name: If the mailbox is on a different street, it must include the street name.
Name height: The name should be at least 1 inch tall.
The USPS will deliver any mail that is addressed to the mailbox although it doesn't have to be accepted.
Any mail that has incomplete information or no name on the mailbox will be returned.