Vid Tjornina (spelling) which means by the lake which is where it is. Very traditional Icelandic cooking with things like puffin bird and cod cheeks. Cozy lounge for afterdinner drinks. Beware, alcohol is expensive and following locals my bring you more too Mcdonalds than to someplace nice.
>>Can anyone make restaurant suggestions?<<
Yes, my suggestion is to bring TONS of money!! Rarely have I felt so poor when traveling - and I'm not a student, but a middle-class professional person. I was there last month, and was SHOCKED at the prices, for food, drink, clothing - everything. I had the cheapest beer at a local bar/restaurant which was called Thorvassen (the place, not the beer) or something like that, in the heart of Reykjavik. It was the equivalent of US$8-9. Also at the same place, I had a bowl of Asian-style chicken noodle soup, a bread basket with olives, and a glass of pineapple juice, and the total came to around $20!!
The top rated restaurant is Siggi Hall. I haven't eaten there because I can't afford to. But it is supposed to be great if you want to splurge. It's a modern take on traditional ingredients
Menu and information here
Here is a list of recommended restaurants by the Tourist Board
I've had very good meals at Apotek and Vid Tjornina. Vegamot is a good choice for lunch, and a great place for late night drinking and dancing.
One of my favorite things in Iceland are the hotdogs bought from the street carts. They are made out of a mixture of lamb and pork. Get them with sweet/spicy mustard, raw onions and fried onions
Hope this isn't too late...
We had two wonderful meals at Sjávarkjallarinn (Seafood Cellar). Creative Icelandic seafood and lamb with a slight Asian influence. I highly recommend the chef's tasting menu - you get to taste everything, and it's better value (for Iceland!!) than ala carte.
I'm a local in Reykjavík. I've had a job in the very center of Reykjavík where most of the travellers will be dining so let me reccommend to you some good spots to have a meal.
Oliver - If the weather is nice you can get a table upstairs on the porch. Order todays special especially if they have fish. Beer is very expensive here! Stick to sodas or water.
Kaffibrennslan - Cozy and rusty. Nice ice lattés. Stick to hamburgers and sandwiches. If you're not very hungry go for coffee and a basket of bread with pesto. Add soup, of which they have some very good ones, and you have a meal! Huge variety of beer - yesterday I had beer 'special of the day' for "only" 350 ISK. That's half of what you pay at Oliver.
Sjávarkjallarinn - an excellent place if you have something to celebrate. For the casual occasion it's very expensive.
Bæjarins Bestu - Hotdog and a small coke for 350 ISK (I think). Local favorite for decades. The sausage is contained with old fashioned sheep intestines. Ask for "eina með öllu" or just one with everything (crispy garlic, brown mustard, ketchup, remoulade and fresh chopped garlic).
Eldsmiðjan - 100m from Hallgrímskirkja. The best pizza in Iceland. Order pepperoni, cream cheese and black pepper. Rumoured for drug trafficking and money laundering.
Subway - What can I say? Order's the month's special if you are on a tight budget. Only 299 ISK.
Vegetarian places are aplenty. I recently had a very nice soup at Garðurinn. They make excellent moisty carrot cake. Further down the street is Á Næstu Grösum. Larger and cozier - also a bit more expensive.
Coffee break - Te & Kaffi on Laugavegur right by the big record store. Very, very good. Excellent salads and cozy surroundings. Go right after lunch when all of the pregnancy vacation parents are there - a good laugh and many cute babies.
Food culture is very internationalized in Iceland. Stick to the trendy places and ask around - people are seriously friendly to tourists. About the only cheap thing in Iceland is the admission price to a thermal swimming pool. Don't miss them.
Tveir Fiskar (Two Fishes) near the harbour would be my first choice http://www.restaurant.is/ - they have a good value lunch menu and have always been excellent on my several visits. Avoiding wine will keep the costs down considerably (that's true almost anywhere in Iceland and Scandinavia)
Although I haven't been to Iceland for a year or so, I'm hoping to visit in October so things may have changed but others I enjoyed were Naust, Siggi Hall at the Odinsve Hotel (where i stayed) and Vid Tjornina as mentioned above which is pretty unique - had some great hallibut there...Also a place above the old town square called Einer Ben was good when I went.
I haven't been back in a few years so can't recommend a specific restaurant (except maybe The Pearl, on top of the power station, for its incredible views), but I do strongly recommend the local lamb at any place that specializes in it. Icelandic lamb is the best I've ever eaten anywhere in the world, and you can't get it in the States.
I just returned from a few days in Reykjavik and will compose individual reviews on our experience with restaurants there. We enjoyed meals at 3 Frakkur, The Sea Baron and the Icelandic Fish and Chip shop. Each were outstanding in their own ways, upon which I will further comment in the posts to follow.
I just want to echo and amplify the comments that have been posted in response to this inquiry: Reykjavik is breathtakingly expensive, in an almost irrational way. I know some will defend the prices: the dependence upon imports, the poor exchange rate of the dollar (which had sunk to 52 IKR during our stay) and the dependence upon tourism of much of the economy. But, as I looked at my wonderful fish at each meal, I asked myself how one could justify the equivalent of $48 US for six ounces of fish, some salad and vegetables. The local beer, Viking, seems a bargain at $6. Coffee is $4 or more at most cafes. Even the Icelandic donuts were $4. There is no excuse to go to the fancy hotel restaurants, including Siggi Hall and the like...unless you're on an expense account. We checked out their menus (as we did for the Seafood Cellar) and most had prix-fixe prices exceeding $100 US. Forget about wine: plonk gets $12 a glass and $80 a bottle. A wonderful bowl of bean soup in the cafeteria at Geysir was $13.
Iceland is a place you've got to experience once in your life, at least. We thoroughly enjoyed ourselves and were astonished at its natural beauty. But be forewarned: it is hard to imagine visiting Iceland and spending less than $700 a couple per day, with most hotel rooms far exceeding $200 a night.