I have a 2009 Mini Cooper and plan on getting wheels with these specs....
Front(16x9.0"/et18/4x100) Rear(16x9.0"/et18/4x100)

What size tires should I get so that it doesn't poke too much? (I'm also putting h&r coilovers on).

 

Well you have a few choices depending on the look you are going for. Obviously with a much wider rim you're going to need a wider tire. your stock 195 55 16 have an overall diameter of approx. 24.44" and you want to stay as close to that as possible while also getting wider. The reason you want to stay as close to the 24.44" as you can is because many of the computer controlled functions of the car such as odometer, speedometer, traction control, ABS, transmission control, engine management all work off the rotational speed of the tire. If you change that too much it will cause all sorts of issues with the car, some minor and some major. In your particular case you want a tire wide enough to fit a 9" rim and as close to 24.44" in diameter as possible. you have 2 choices. a 225 45 16 or a 245 45 16. Neither are exactly 24.44" but they are both as close as you an get with what is available. the 225 45 16 is slightly smaller at 23.97" and a little narrow for a 9" wheel but it will work. and the 245 45 16 is slightly larger at 24.68" and plenty wide for a 9" rim. I'm sure your question would be "why not split the difference and do a 235 45 16?" which would be almost perfect at 24.33". the reason is to my knowledge that tire size does not exist from any manufacturer.

All that being said in both sizes there are not many to choose from and of the ones that are available in the 245 45 16 most are expensive race tires and would not be suitable for daily driving simply because they are going to wear out very quick. Of the 225 45 16s that are available most are also race tires but there are a few to choose from in the 225 45 16 that are not. Just keep in mind that the 225 45 16 will be a little bit stretched on a 9" wide wheel. Hope this helps. Good luck with your project.