In an effort to make iOS 7 more simple and clean, Apple decided to make the Messages app only show a person’s first name. Unfortunately, I have sent messages to the wrong person because I know many Davids, Tommys, and Marks. You are, however, provided an option to always show a person’s full name.
Visit the Settings app > Mail, Contacts, Calendars > Short Name (in the Contacts section). Turn off Short Name and Probably “Prefer Nicknames”. If you leave Nicknames on, you will not see their last name, either.