There are kind-of three different things you mention there:
You could actually generate this and add it in. Either with a high-gain op-amp like Paul suggested or a single-transistor noise source.
This is the hard one, I reckon. Some random staticky clicks and pops isn't going to be easy to fake. If I was doing it, I'd program up a random static generator on a PIC, but I can't imagine a *simple* way with analog.
3) Lo-fi sound
You can get this by limiting the frequency range of the guitar with filters, so that you get that "on a cheap radio/on the telephone" sound. Coupled with running it through the cheapest, nastiest old transistor you can find from a broken toy, you'll be pretty close.
Perhaps you could achieve both (1) and (2) by getting one of those single-chip FM receivers and then running it not-tuned and/or without an arial. That'd get some interesting random noise into the circuit.