For the more than a couple trips I've hammock-camped, I have needed a little bit of rain cover at night. There are several options for a nylon tarp set-up or ENO's actual hammock tarp, but I wanted something on the cheaper side. I've got a nice enough tent that I would just pack if I anticipated heavy rain. But a tarp option that packed down small and didn't add much weight would be perfect.
I cut it down to 9'x9' and bordered all the edges with Tyvek tape to keep it from unraveling. If you do your own Tyvek project, be sure to invest in a roll of the tape. The cut edges do eventually fray. But the tape will prevent this and keep a nice edge around the whole piece. I debated on whether or not to install grommets. But I opted to just purchase tarp clamps from Dick's that will allow me to attach my lines to any spot on the tarp I'd like. That'll come in handy when I find my hammock perched in a precarious spot and cant' figure out where to tie it all down.
If you're looking for a cheap, light and minimal tarp or ground cover. I'd go for a homemade solution like this. It seems to be simple enough.
Get out there.