It'd be a viral winner (puppy mode vs cat mode!), but how often do people really need this sort of function? And when they have the problem, how many people just email the client "What browser are you using?" (initiating that "Is it Windows?" conversation) rather than Google "best way to get support information from my nutbag client"?
Cat mode sounds immature, but MailChimp gets by.
Is there another common process where by technical people need to collect information from internet battlers?
"What browser are you using?" is probably one of the most commonly used methods, and also the most pointless question that one could ask of a non-tech-savvy client. You're bound to get replies like "I'm using Google" or "What's a browser?"
A "cat mode" that gets widely noticed could help many of us deal with clueless clients. Make the cat follow laser pointers and mutter lolcode while it collects every possible piece of information about the client's system. Really, anything to make it more viral would contribute to the Greater Good (tm) and more than compensate for immaturity.
Even something as simple as a browser-specific tuturial for clearing the cache would be immensely helpful. For example, Dropbox customizes its download & installation tutorial for each user's browser. It's really helpful.