What does it mean 'name you go by'?

Answer 1

In American English, go by can definitely refer to name. What do you like to go by or what do you go by sound fine to me, in addition to Forero's suggestions. All of them mean what name do you prefer to be called? However, the original question of family name vs given name seems odd to me. Usually, what do you go by would refer to a choice between a given name and a nick name (Do you go by William or Will?), or between two possible surnames (Do you always go by your married name, or do you still use your maiden name professionally? Do you go by Sanchez Martinez, or just Sanchez?). It doesn't mean what do you want to be called in this particular situation but generally what name do you use for yourself? Except for some (mostly young) people who use their last name as a nickname, most people don't go by only a given name or family name -- they go by both, one, or the other depending on the situation. So it wouldn't really make sense to ask do you go by your family name or your given name?

Answer 2

The "name you go by" is the name you are most commonly called. For example, your name is John Joseph Smith, the name you go by would be the one most commonly used. Perhaps Joseph, Joe, John, Johnny, etc.