If you're traveling for more than a day or two you should bring your own mountain bike and have some touring experience. Outside Reykjavik bicycle repair shops are few and far between and parts can be hard to come by. You should be able to fix flats, change brake shoes and have the tools to fasten every screw, nut and bolt that might come loose. You should bring extra patches and tubes, two sets of brake shoes, spokes with nipples, derailleur- and brake cables, oil, chain links and a chain key. And don't forget a first-aid kit for yourself and a sowing kit for your clothes. You should consider giving your bike an overhaul before leaving home, greasing up the hub bearing, pedals and bottom bracket, etc. and use it for at least a few days afterwards to see if any problems pop up.
One common problem people encounter when cycling in Iceland is a broken rack. Aluminum racks break easily on rough roads and only strong reinforced racks are recommended You should have front and back racks and try to distribute the weight 50/50 or 40/60. If you load everything on a rack in the back the stability and steering is compromised, especially on loose dirt roads and going uphill - you don't want an accident. Do not carry weight on your back.
You can encounter all sorts of weather so waterproof panniers, such as those made by Ortlieb,Vaude and Karrimore etc., are great for keeping your gear and food dry. These also have quick-lock systems that lock the panniers to the rack, preventing them from falling off every time you hit a bump and maybe causing accidents. The tent should be a four-season tent, able to withstand heavy winds and the sleeping bag should be able to keep you comfortable down to freezing point. A good mattress, such as "Therm-a-rest ultralight" or cheap foam one, is necessary to isolate you from the cold ground, especially if you're using a down sleeping bag.