Well, both of you are correct. In general terms, a wider tire has a greater contact patch with the ground, so can provide traction. As your friend stated, though, the tread pattern/depth will have a lot to do with how the tire performs during inclement whether.
Take for instance a race tire which is rated at a width of 325mm. With your line of logic, this tire would provide excellent traction for the vehicle. And this would be true, unless the tire was driven on the street where there is water. The car would not have the traction needed to sustain type of safe operation.
Take the same 325mm tire and place it on snow. You can expect a loss of traction. In fact, a skinnier tire will work better in snow than a wide tire would. The reason for this (I believe ... no empirical evidence) is because it has more weight per square inch due to the smaller contact patch. It also cuts through to the ground better instead of riding on top of compacted snow.
There are other factors involved here as well. If a tire is made to last longer (say made of harder rubber), it may not have as good of traction as a tire of the same width and softer material. Tread patterns themselves have a play in traction. Side wall height, tire flex, and inflation also have a play in it.
Another area to consider is what is the physical dimensions of a tire will you be able to fit under your vehicle? There is a trade-off here as well. Another trade-off is cost, the wider the tire, the more expensive it will be (all other things being equal).
Unfortunately, there are so many variables when considering traction, you just cannot put a generalized statement upon a single given factor, which is width in your case. To provide the best tire for your application takes research, bringing all of the factors together to determine your best course.