As a child, we went to Hilton Head almost annually. I haven't been for many years, but hope to go back this summer. Anyhow, there hasn't been much posting on this thread yet, so I figure I might as well provide some suggestions, although my experiences are a bit dated.
My parents used to leave the kids at the hotel while they had their favorite adult dinner at Alexander's. They used to get lots of business, and it's nestled in one of the plantations.
http://www.alexandersrestaurant.com/
I enjoyed a very nice meal at Old Oyster Factory in more recent memory. In going to their website, I just found out that this is a sister restaurant of Alexanders. This restaurant was featured on national TV (Food Network, I think) because they have a great sunset, so try and go for that.
http://www.oldoysterfactory.com/
The above restaurants have yet another sister according to the Old Oyster website. This seems like a newer, more contemporary option.
http://www.redfishofhiltonhead.com/index2.html
There is a popular bakery there, Signey's. I remember it having key lime stuff and also cafe lunch stuff.
Maybe check into these. Plus, if you're staying at the Westin, the concierge there should be able to confirm which suggestions are best.
Although off topic, if you have time, I would highly recommend renting bikes, which are really fun to ride along the ocean. Also, I've been on a great dolphin cruise that was very entertaining and not commercial.
Please report back!!!