I'm guessing January isn't the waitress' name. Perhaps the customer meant that it felt like a month had passed while she/he waited to be served.

I've only recently learned the slang use of "January." January is not the waitress' name. Because January is the first month of the year, it is associated with new beginnings, etc. Calling someone "January" is somewhat of an insult, as in it's obvious that they're new/inexperienced/etc. When they said "good work, January," they were mocking her obvious inexperience.