There are quite a few restaurants available on our sleepy little island. Here are some suggestions and rankings from Yelp
Tribuzio's Grill at Jekyll Island Golf Club - The restaurant at the club. Looks the same, but they added Cajun Crawdad Biscuits & Gravy which is a red gravy and has a nice spice to it.
Jekyll & Tide - this is a remake of the Driftwood Bistro at Villas by the Sea. It has changed quite a bit. Very pricey, higher-end meals now, few options below $20 in the main restaurant. The bar is much nicer and now has an outdoor patio by the pool. When I visited there this September, they did not show me a different menu for the bar but one for the tavern is listed on the website. Also, they did not offer breakfast when I was there. If there truly is a separate menu for the bar side, still might be a viable option because of convenience.
Love Shack BBQ Shrimp and Yardbird-- one of my favorites, pretty quick service at the counter and has good BBQ and fried shrimp. It is located at directly after the traffic circle when you take 520 on to the island near the Westin
The Wharf At Jekyll Island Club Resort - The Wharf has excellent shrimp baskets. Pleasant quick service. Honestly, I think some of the best fried shrimp I have ever had
Zachary's Riverhouse--This is usually ranked the best place and is located on the intercostal waterway on the right as you come into the island
Takeout
Red Bug Motor's Pizza -- I am pretty sure others have used it. Located near the miniature golf course http://www.redbugmotorspizza.com/