Hi nmclimber,
I spent a month in Iceland a couple summers ago and did a fair amount of hiking. With just ~7 days, you'll probably want to stay in the southwestern corner of the island, otherwise you'll spend all your time driving to/from TH's.
A few areas/hikes come to mind:
1. Just north of Reykjavik is an excellent dayhike to a waterfall called Glymur. The hike is not too demanding, but is very scenic and not crowded at all. The beginning of the hike passes through some cool lava tube/cave formations, then ascends to the plateau above an immensely deep and impressive ravine. Views at the top are awesome, and peering down into the ravine is dizzying. I had a great time hiking there! The info at this link seems good:
http://www.go-to-iceland.com/glymur.html
2. The hiking around Landmannalaugar is fantastic and stunning, though the drive out there might be prohibitive for you (unless you spend your entire time at Landmannalaugar). I didn't do a specific trek, rather I just spent a few days exploring the area in a series of dayhikes, of which there are dozens. Not sure about the huts beyond the main one in Landmannalaugar, which was quite nice. I'm petty sure I saw tents, but I know that most people prefer
not to tent because of the fickle weather. Another option is sleeping in your car (which you will certainly need to get around to the hiking areas in Iceland).
3. There's a great dayhike (possibly an overnighter?) following the river/falls above Skogafoss on the southern part of the Ring Road. The views up to Eyafjatlajokull (the one that blew a few years ago) are beautiful, and there are too many waterfalls to take in.
http://www.world-of-waterfalls.com/icel ... afoss.html
All the other hikes I did were in the further reaches of Iceland, so I don't think you'll have time for them. Hope this helps!