Manarat Al Saadiyat is an art museum I try to visit every now and then. I often find interesting and inspiring local and international exhibitions here. The museum has multiple gallery spaces and is more of a cultural hub than just a venue for galleries. The adjacent Fanr restaurant with its outdoor garden is definitely worth a visit to have your lunch or coffee and deserts.
It's located on Saadiyat Island, Abu Dhabi. 

Photo credit: Manarat Al Saadiyat

Photo credit: Fanr Restaurant

Al Fanar Restaurant offers local Emirati cuisine and the restaurant's atmosphere is designed to capture the feeling of traditional local culture. The name "Al Fanar" means "lantern" (often of the color blue). There are two Al Fanar restaurant in Dubai and one in Abu Dhabi. Many of the dishes are delicious and many of my local friends enjoy a lunch or dinner here on weekends. 

After a 15min short ride with a speed boat from Saadiyat Island (Abu Dhabi), you will reach Nurai Island. Zaya Nurai Island provides transportation, lunch and access to the beach and pool for 350-550AED depending on what package you buy. There are also pricey villas to rent if you wish to stay the night. We, however, enjoyed our full-day by the beach and at the restaurant. 

Café Arabia is a great alternative for breakfast, lunch or coffee break. I come here every now and then for a light lunch and read the newspapers that they have laying around on the tables. They also have book-swap-shelves. If the weather allows (if it's not too hot), I recommend sitting on their terrace.
Google Maps show a slight wrong address (wrong by one street) - Café Arabia is located on15th Street, "Mohammed bin Khalifa Street" correct map (and not on the backstreet).