We are not married so I have a different surname. The baby is getting my boyfriend's surname because I think it's simpler than a hyphenated surname. (both our surnames require us to spell them for people all the time, so I am all about making life simpler in regard to names)
However my boyfriend is a different ethnicity than I am, and if I were to travel alone with our son, I don't want people to think I abducted him. I figure if my crazy last name is one of his middle names on his passport, then no one can deny that we have a family connection.
I don't have any plans to change my name if we were to get married.
ETA: It seems that to travel, I would need a letter from Dad, indicating that we are travelling with his knowledge and permission, and it would help to have a long-form birth certificate, which would include my full name. So including my surname is probably not necessary anyhow.