November is kind of rough for Alpine stuff, probably still not enough snow on glaciated terrain to close up the crevasses completely and very short days / daylight.
You might get good ice climbing conditions in November, there’s almost a lifetimes worth of climbing around Reykjavik. You can see all the topos on this page under Crags or routes, should be able to figure it out using Google translate.
There is also plenty of mountaineering that would be fun in November. I would actually just suggest you rent the cabin that ISALP has in Tindfjöll (Summit Mountains) wich is located in the south. There is plenty of mountaineering accessible from the cabin, and some of it involves crossing a small glacier, Tindfjallajökull. Info about the cabin is also here on the page.
Finally, the outlet glaciers are fun and it’s easy to set up a top rope for some easy ice climbing. That will be nice in November. If you are not used to navigating that kind of terrain there are plenty of guided tours on Sólheimajökull glacier in the south.
Have fun,
Sissi