There are many metrics of what "lightweight" can possibly mean: disk size on server, amount of code generated for the client, difficulty of installing / configuring process on the server, etc...
I've been on the core development team of a PHP Framework called NOLOH for 6 years now, and I believe that NOLOH scores remarkably high on all those meanings of lightweight. NOLOH's benefits are too numerous to just quickly review here, but we believe that it makes developing and maintaining a website or webapp of any size or complexity orders of magnitude simpler than other solutions.
Yet what distinguishes NOLOH, as far as "lightweight" is concerned, is that NOLOH will automatically figure out what is and is not used by the particular project, and lazy-load only the necessary modules, while also sending the client only the necessary code for his current view only. We describe this aspect as being "on-demand." Many other frameworks either include far too much (commonly found in fat client frameworks) or leave it to the developer to surgically include what he needs. NOLOH handles this, and countless other hassles, for you completely transparently and automatically. In short, it lets you just focus on the application logic without the countless implementation tricks and details.
Check out http://www.noloh.com