Hawai'i is the name of the 'big island' - the name 'big island' is not it's official designation
I agree with the other posts - during busy holiday periods, avoid the typical destination islands like O'ahu and Maui and try something slightly different like Kaua'i and Lana'i - although the latter has fewer accommodation options, and thus could be quite pricey around the holiday time frame.
Personally, I like the big island - lots to see and do, spread out to avoid clutches of crowds and busy traffic, and offers a number of diverse micro-climates. On a full week trip (arrive Sat am, depart following Sat or Sun pm), I'd do a few days on Kaua'i and a few days on the BI.