There is no island in the Hawaiian chain called, "Pua." The show that the Tanners stumble upon is a show at the Polynesian Cultural Center on Oahu which is Hawaii's number one paid attraction.
When the Natives come upon the Tanners, one tells "Mele Kalikimaka." That's Hawaiian for "Merry Christmas."
Aired on the same night as Family Matters makes it's debut on ABC.
Northwest Airlines, who provided transport for the "Full House" team (as mentioned at the end of the episode) was purchased by Delta Airlines in 2008. The name Northwest Airlines ceased to exist on January 31st, 2010.
The Turtle Bay Hilton & Country Club provided filming of other shows and movies like Hawaii 5-0, Magnum P.I., Cougar Town and the short-lived Fox TV series North Shore. The resort featured as the setting for the reunion-movie Baywatch: Hawaiian Wedding and for the 2008 film Forgetting Sarah Marshall.