Although it is possible that it was used at the San Diego "El Patio", I think the vast majority of evidence points toward it having been used at the Cairo, Illinois "El Patio".

Evidence in favor of Illinois;
a.) the chip has a similar hot stamp to the 1$ and .50 cent T-mold Cairo chip, for which we have the order card.
http://cgcm.themogh.org/cg_chip2.php?id=ILMCEP&v=630527733 (scroll to the bottom for the cards).

b.) as Virgil points out, the hot stamp is also similar in style to the the Cairo "Frog City" AroDie chip, owned by the same person.
Frog City Club: http://cgcm.themogh.org/cg_chip2.php?id=ilcafc

c.) It fits into the color scheme of the Cairo chips, the AroDie $1 chip is the same color as the $1 -T-mold Cairo chip, for which we have an order card.

d.) as Doug points out, it appears that Gene Trimble wrote that Ed Hertel found the $1 AroDie and $1 T-mold Cairo chips together, and got them from an heir of a person that worked at the Cairo "El Patio" club.
http://www.marlowcasinochips.com/links/genetrimble/illegaloftheday/750ClubIL.pdf

e.) The Cairo club opened in 1948, which is about when we believe that Homer N. Garrison (who later owned the Reno Game Supply Co.) first started using the AroDie mold. Gene Trimble wrote a history on the AroDie mold; http://www.ccgtcc-ccn.com/Arrowdie.pdf (The Phoenix chip he pictures at the end of the article has been dated to 1947.)

The only evidence in favor of California is that the San Diego "El Patio", was a licensed card room, open from 1950 to 1953. So the place definitely had chips, and the AroDie does show up in California during the early 1950s.
San Diego "El Patio"; http://cgcm.themogh.org/cg_chip2.php?id=CASDEP&v=1296559981

Unless anyone objects, I will remove the AroDie chip from San Diego, but add a note in the listing explaining it was previously attributed to San Diego. I will also add a hyperlink there to the Cairo listing.