Yes, it is true however more important is getting the right rubber for your needs.

If you are doing basic street driving and using a tire that you want to wear well and no track then a light tire like General Exclaim UHP would be worth a look. It's lighter for a given tire size but it's not the best handling tire for cornering or performance driving.

If you want the best performance then you can't be too concerned about a little more weight traded off for better tire design and materials and better size for your intended use. Many times for track or autocross we use tires that are wider to get more tread but we do add weight. In addition to having wider heavier tires we have more negative camber to make better contact with the ground and enough horsepower to make everything work.

Using a stock MCS or base Cooper with limited power you'd want to keep things reasonably sized (not too wide) and fairly light.

Right now I have basic Holey 15" wheels and stock Conti 175/65-15 tires on for daily street driving with my autocross ready MCS and it works much better than when I have my 17" R90 wheels and runflats on (for looks). I can't take corners quite as aggressively but for everything else it's good enough and much more comfortable.

 

I saved $200 by switching to Geico, and 5 lbs. per wheel/tire by switching to Black Konig Feathers with Falken 912s (215/50R16) for my 2007 MCS (great price on wheels from Altamini - list is $139 and 4 tires shipped for $276 from DT).

I made the swap for my Bridgespokes with runflats right away so cannot speak to the difference - but I think the weight loss (intertia reduction) would make MOST difference in stop and start driving and or accelerating from low speeds around tight corners, as this is where it is most difficult to overcome intertia - once you are moving, overcoming intertia is not such a big problem.

(just kidding about Geico - no Geico in MA

 

Lighter wheels will make the MOST difference in handling over bumpy roads.

 

This is my general approach. . . I call the Contis my commuting tires. Plenty zippy around town but not as secure on the open, open road. Since most of my driving is mundane on my commute I feel good about not burning expensive rubber for no reason.

 

You can even commute in style-

Use the stock 15" wheels

Use 205/55-15 tires in summer or all season type for comfort and longer treadlife.

Use 205/50-15 or even 195/50-15 performance tires for more fun. These are smaller than stock tire diameter but some good tires come in these sizes.

Check the load rating and look for at least 84 if you are going to put more than 2 people in your MINI.

Good thing is that many good tires are relatively well priced in these sizes.