EDIT: I need this for my custom web part.
If it's your custom web part, you could just output the web part properties in the render method. Either you directly render java script code, that sets some variables or you can output some html that you can access with your client side code.
Have a look at the source code for SharePoint LiveListData. I am reading the ListName property from ListViewWebParts on the page using jQuery and the Web Services.