Based in one hotel for 1 week? If so, where?

If you want to stay in one place the whole week, Reykjavik is the obvious place as you can visit (part) of the south coast, part of the southwest and Reykjanes + those tings in and around Reykjavik (Gullfoss, Geysir,..).
If you do not want to do the usual drive round the island, you could consider to spend some days in Reykjavik and then take a domestic plane to Akureyri, rent a new car and see, Myvantn, Krafla, Dettifoss etc. and the fly back. Domestic flights are cheap and depart/arrive at the very convenient Reykajvik airport more or less in the middle of the city. Only probelm is that a week is a bit short for that.

Note that iceland is wet all year, but October is the wettest month.

Hi Bjoern,
Thank you so much for your reply. We just booked tickets and will be arriving on Sept. 28 and leaving Oct. 5. We decided to stay in Reykjavik for the whole time period. We'll be renting a car and probably doing one day-flight to the north. I realize the solar activity isn't the best for northern lights activity now, but if they were to flare up, would it be too bright to see them in Reykjavik?
Thank you!
Liz

"would it be too bright to see them in Reykjavik?" - depends on actual intensity at the occation - but proabbly yes.

Since you have a rental car, and Reykjavik is a reasonably small city, you just stay sober and take a drive outside Reykjavik (towards North - which in reality will be East - but not south) to et to a dark place with citylights in the back/at the side looking North). You watch for as long as you want and go back to hotel).