Tasha: I think it's not "Bético" but Betico (the accent is on "ti") and probably comes from the nick name "Beto", from Alberto.
In Spanish there are some "little part of words" that we can put at the end of a word, attached to the root and in grammar these parts are called "sufijos". There are some "sufijos" that have the meaning of "little". Examples:
armario: armarito (pequeño armario) lleva el sufijo "ito"
mesa: mesilla (pequeña mesa) lleva el sufijo "illa"
perro: perrico (perro pequeño) lleva el sufijo "ico"
The sufijos sometimes can be used meaning "dear and little", and it seems it's the use in "Betico, it's like saying "little Beto" but with love.