Yea. AE or Motion will be your best friend here for getting a real nice, stylized look, but it should be possible in FCP if those tools aren't available. I'm really not a huge fan of FCP's noise generator though, it looks too fine and generic.
I once achieved a nice effect by recording TV static, and playing it back through a broadcast monitor. There was no sync signal on the tape, so the image flickered and distorted as it was going into FCP. I then played with opacity and composite modes to blend it with the footage that was to sit under it.
If you want to distort the video that sits underneath it, consider finding some static that has a nice black/white contrast, and making a displacement map out of it.
good luck.
Bryan