The name of the island was Vella Lavella in the Solomon Islands of the southern Pacific.
Source(s):
http://en.wikipedia.org/wiki/Baa_Baa_Black_Sheep_(TV_series)#Production_not...

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From the wikipedia page about the series:

The name of the island where the real-life Black Sheep were stationed was Vella Lavella in the Solomon Islands of the southern Pacific, known as The Slot; this was changed to Vella la Cava in the series for legal reasons (the same was done for the Rear Area Allied Command island of "Espiritu Santo" (Espritos Marcos). The Channel Islands off the coast of Southern California provided an adequate substitute backdrop for flying scenes. Filming of airfield scenes was primarily done at the now-closed Indian Dunes Airport in Valencia, California. Occasionally, modern trucks can be seen passing in the background.
Source(s):
http://en.wikipedia.org/wiki/Baa_Baa_Black_Sheep_(TV_series)

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Wikipedia states that it was the following:
Russell Islands; New Georgia and Bougainville; New Britain; New Ireland
Source(s):
http://en.wikipedia.org/wiki/Pappy_Boyington

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Vella Lavella was where the real black sheep were.
In the series they changed it to Vella La Cava
Source(s):
http://en.wikipedia.org/wiki/Baa_Baa_Black_Sheep_(TV_series)

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