What does "basis" mean in "first-come basis"?

It means policy.

See the first definition here: http://www.macmillandictionary.com/dictionary/american/basis

A policy is more or less an adopted system.
http://en.wikipedia.org/wiki/First-come,_first-served

Taking it a bit farther, the mental picture would be of a queue (line of people) where (in the absence of a number-assignment) the first person to arrive will be the first person served/acknowledged/chosen.
Expanding the topic to a similar sounding expression, "on a first-name basis" is an established friendship that permits each to call the other (person) by his/her first name to avoid some