Question: My friend and I are visiting for 5 days in early Jan and are uncertain as to whether we should base ourselves in Reykjavik and do multiple single day tours. Or go on an actual multi day tour like this one. 
<https://www.adventures.is/iceland/multiday-tours/multiday-adventures/golden-circle-south-coast-ice-cave/>

Can you someone give me an idea of the pros and cons of both options?

The main priorities are seeing the northern lights, glaciers and ice caving.

Answer: 
I spent 10 days in Iceland in march. 

Pros of the Tour: Everything is settled! This is vacation for you, you do not need to organize all the stuff. You get your tours, maybe picked up at your hotel and do the tour. Don't waste your time on organizing stuff and thinking about how when an what to do. However, everything is already planned. You are not flexible at all

Pros of single day Tours: Your are more flexible. Feeling ill, weather is bad, you are totally in the mood of doing XYZ instead of the "scheduled" thing? No problem, those single day organized tours are more flexible. However, if you take tours flexibility is limited. You also need to book in advance, especially in January not every tour will be every day.

Finally I want to introduce you a third option: Do most of the stuff on your own. Rent a car and do it (things like Jökulsárlón Glacier, Golden Glacier etc. are easily done alone with a car). If you like exploring on your own, being your own boss organizing stuff on your own etc. this might be a good option. Just because travelling is (for some persons, e.g. me) also exploring and discovering of the "unknown" (unknown for yourself). If you are shuttled everywhere there is not much discovering any more. Book the things you cannot do alone (ice cave) and do the rest on your own. However, it is more time consuming, you need to organize your vacation etc.

Probably its all about your preferences: You want to relax or is organizing no problem? You want to see e.g. the northern lights really badly? 
Highest chance for northern lights is actually on your own. Because you can go whenever you want with your car on the ring road and go to a dark spot where you can see them
A multi-day tour will have access to further locations. There are still plenty of things to do within reach of Reykjavik but if you want to see Jokulsarlon, which I would say is definitely one of the beautiful spots in Iceland, a day is too short to reach it and back.

The advantage of day tours is that you have your own choice of lodging and you do not have to move your stuff between places. It's a different pace but also a you have a different reach. You also have more flexibility for meals.

Note that most of the day is dark in Iceland in January, so even with several full days of tours, there will not be much more time for site-seeing than with a single-day tour. Now, if this were June, you could be site-seeing in the middle of the night. Actually, the light was so beautiful that two nights I decided not to sleep at all in order to take more in!