MICHEL: There is a man with a funny accent on the phone asking for you.
LORELAI: Really? Did you guys exchange the secret handshake?
A secret handshake is any non-standard handshake used to signify fellowship within a certain group. The most famous organisation with a secret handshake is The Freemasons.
It is not clear who the man with the funny accent is: it is presumably either Mr. Medina or Headmaster Charleston, neither of whom has an especially hilarious accent. Maybe all Americans sound funny to Michel.
The Annotated Gilmore Girls 