I've set a break point that gets hit on an ajax call correctly, but as I'm trying to debug it, the debug session automatically ends when the client times out waiting for the ajax call to return. The console in the browser says:

Failed to load resource: the server responded with a status of 500 (Internal Server Error)

Is there anyway to continue the debug session after the client times out? Or how should I solve this?

First of all, try to set browser timeout to higher value.
If you installed php as CGI or FastCGI, you have to update timeout on the server too.
Finally, if you run php under Apache, read this http://www.php.net/manual/en/features.connection-handling.php