For starters, there's a phrase which some people use that's entirely unique to these islands. Since the Hawaiian alphabet, consisting of seven consonants and five vowels, does not contain the letters r, y, c, s, or t, the way they say Merry Christmas is Mele Kalikimaka. There's even a carol for it.
Also, lights are a really big thing here. The city of Honolulu, private homes-- everything is covered in tiny flashing bulbs. I think the mentality is "if there isn't going to be snow, there'll at least be lights;" so even though you're wearing shorts, you still know what time of year it is.
The most popular Christmas song is probably the Beach Boys' "Christmas Comes This Time Each Year," and the meaning of "White Christmas" for us in Hawaii is more as a descriptor of the foamy water created by the colossal surf.
Then there's the typical ways that people in Hawaii like to celebrate all holidays: family camp-outs, barbecues, and fireworks. Whenever there's time off from work, that means it's time to gather all your uncles and aunties and pitch some tents on the beach; it's not chestnuts, but rather, hot dogs and burgers that we smell roasting on open fires; and come New Year's, the smoke from all the fireworks is so thick that it can make driving hazardous, and public service announcements suggest closing the glass windows of children's bedrooms (which is probably the only time they're closed). In other words, people really like their fireworks.
So even though we are wearing bathing suits and toting surf boards, it does not mean the holiday sentiment was lost somewhere in the Pacific.
Aloha, Mele Kalikimaka, Happy Hanukkah and best wishes for the New Year.
