You need a framework on the server side, there are good python options: Django, Flask, Bottle, webpy. That framework will receive a (ajax) request process the GeoJson and returns the string.
After the information enters the framework you are now "pythonic" and can use the tool of you preference to read and work with the GeoJSON, that could be for example Shapely or Osgeo OGR Python bindings. If you choose Django, it already come with it's geometry helpers - GeoDjango.
On the client side, you need something to make the steeps of making the request, receiving the answer and showing the info box easier. For that you could use frameworks/tools like ExtJS, Jquery (UI), Dojo, KendoUI
Personally I use a combination of Django + ExtJS to accomplish this kind of thing. It's a heavy load combination, I mean, it's has far more resources than you need. But is so easy to use and it's so very well documented that I wouldn't recommend other thing to get started. After doing the basic tutorial of both (Django, Extjs) you have enough information to accomplish the task stated on the question. Later you can try other options.
Btw, Extjs if free only for open source projects, take that into account.
Btw2, to run the python framework on the server you will need a web server with wsgi. My option is Nginx+uwsgi. Take a look at the deploy section of django documentation.