Silverlight Client:
Silverlight has several advantages including UI flexibility, greater WCF control and being able the same programming language/tools on client and server. If you chose Silverlight you have more flexibility over how data is transferred to and from the server, such as what protocol to use (TCP, HTML) and how to transfer data. The UI options with Silverlight allow you create a very rich and compelling experience with the complete designer support and animations.
ASP.NET Client:
A simple HTML client may be faster to create since you would not need to learn how to use Silverlight to create the equivalent functionality. Even with just HTML, css and javascript, you can still have a very nice user experience just look at gtalk as an example. Using modern javascript libraries like jQuery make it even easier to create a complicated application quickly that is still easy to understand and maintain.
If you are looking to play around, have some fun and learn something new perhaps consider using a ASP.NET client with a basic WCF RESTful interface. You will still have alot of control and flexibily on both the client and server regarding what messages to send, how the client makes requests, caching on the server, etc. For an easy to digest post regarding RESTful WCF check out this post by Rick Strahl.
I wrote a similar type of application to perform notifications within an ASP.NET application and was waffleing between doing push vs. polling from the client (see this stack overflow question). I was inspired by this chat example which used a very simple interface and push notification for the client.