I have the R120 4-hole circular spoke wheels. Besides being part of the ET48 family of wheels and being 16x6.5 I cannot find any more info on them and I was looking for the weight of the wheel. I have not seen many of this style so I like that about this wheel. I am about 5k miles from needing tires and I am starting to do my research on tires and wheels. I am planning on ditching the run-flat tires but what are the advantages of a different size wheel over what I currently have?
Image shamelessly stolen from http://minicooper.wikia.com/wiki/OEM_Wheels_and_Tires

 

I think those are around 20lbs or so. If you really wanted to find out just take a wheel off and weigh it. Tire rack has all the weights for every size and model of tire so just find your tires weight and subtract that.

Bigger means more expensive wheels and tires and worse ride quality, but it also gives many people the look they are after. Vice versa for a smaller wheel. For instance, my buddy has 17s and runflats on his stock R53 and it rides comparably harsh to my car with h&r springs and koni yellows because I have 16s with non runflats.

If by different size you meant wider and a lower offset, you could always just get some spacers for your current wheels, as you mentioned that you liked their uniqueness. Although you'd still be where you were regarding weight.

 

The R120 is a standard MINI 16" wheel made by BBS for MINI and is cast alloy. It's weight is likely 19-21 lbs which is a little heavy but built that way to give you a little strength for street use over bad roads.

The most common upgrade is just to move from runflats to non runflats which allows you to change tire size (although some tire shops will complain, the following sizes are fine for OEM 16" rims):
195/55-16
205/50-16
205/55-16

Tire size alone can change the sidewall height and affect the ride quality, a 50 series sidewall will be stiffer and firmer riding than the same tire in 55 series. There are even wider tire sizes with shorter sidewalls that can be used with 16" wheels for track use.

Ride comfort and noise will be improved by switching to runflats alone, which tire you choose will also affect noise and comfort as well as handling and treadwear. No tire is perfect.

What are your roads like? Smooth as glass or full of potholes? What suspension are you using? Any plans to lower it? Will the tires need to work in cold weather or any snow or do you have winter tires? Any use other than street driving? How do you drive- aggressively or just a little sporty.

What are your needs for tire use, you need to prioritize in order to reduce the choices-
Handling
Comfort
noise
wet vs dry performance
treadwear
Use in all seasons and cold weather
Price.

Budget is a big deal and more so if you want to replace wheels and tires.

Moving from 16" wheels to wider sizes but still in 16" is common. 16x7 or even 16x8" is possible but wider usually means heavier once you go beyond 7" wide wheels. The same tire sizes will fit. Of you can also go wider but the tires will weigh more.

Moving to 17" wheels is fine but tire sizes will be more like 205/45-17 and 215/45-17 which means the tire sidewalls are stiffer and firmer riding. If you are only commuting then 16" are more forgiving- same if you have bad roads.

So provide more information when you can.