Your best bet would be to get a good widget app and then you can configure a very small 1x1 widget just for launching the weather. I have tried many weather apps and all have their plusses and minuses, but I never really looked at which each had for their own widgets. I'm currently using Accuweather which has a 4x1 and 4x2 widget which is bigger than what you want.
I use HD Widgets and have a large 5x4 widget that gives me time, current weather, date, 5 day weather and when I press each of these areas it launches a different program. But again, that's not what you are looking for. Any widget app will able to program you a small widget and launch the weather program of your choice. Tons of posts on weather programs and widgets. Do a search, grab a beverage and settle down for a couple hours of reading!