Well, any problem that has a wide range of possible solutions which can be discovered and optimized faster by an AI algorithm than a human...
I'll just throw some random ideas out there:
Classifying users based on user behaviours (time spent on pages, links clicked, mouse positions, all sorts of other measurable facets of user behaviour). You might want to classify them in all sorts of dimensions, the possibilities are basically endless.
Interactive simulations with agents driven using neural networks and genetic algorithms.
Classification of user drawn images.
Sentiment analysis.
You can do whatever AI stuff you want on client side. The question you should ask is when to do it client side and when to do it server side. Obviously if you want to keep your algorithms safe then don't write them in Javascript that can be viewed client side. If you need to have reliable performance, don't do it client side.