There are a few things to think about:
Fabric. Selection of fabric will affect weight and durability. Whatever you get, make sure it's UV resistant, and you should probably consider something with rip-stop. For me, the ultrasil nylon is a bit light (I doubt it would handle much of a tree branch falling on it). At the same time, a durable canvas tarp would be relatively heavy. Mine is a 210T rip-stop nylon -- not the lightest, but it's plenty durable.
Dimensions. Too small is useless, too large is extra weight. I use a 10x10 foot tarp, which is quite spacious. In fair weather, it is plenty large (possibly too large!) for my hammock. In the rain, I have plenty of room to relax and cook beneath before stringing up my hammock. I'd consider an 8x8 an absolute minimum.
Weight. Obviously, this is a concern if you're thinking about a tarp, but it's a trade-off between fabric and size. You'll have to make your own call on what's acceptable.
Construction. This will not add much to weight, but may add a great deal to both overall quality and price. Check the seams on the edges and how the grommets/tie-outs are attached - a brisk wind could easily tear these. Seams should be seam-sealed. Compare a few vendors product, and you'll begin to learn what works and what does not. Find someone who knows something about sewing and have them look (my wife sews professionally).
Positioning of the tie-outs or grommets. This will affect how your tarp can be set up. Because of the positioning of attachment points, some tarps must be set up either square or on a diagonal. If this is important to you, be sure you check first!! Ideally, you should have loops which allow the ridge line to be run above the peak of the tarp rather than pitching the tarp over the ridge line. Allowing the ridge line beneath the tarp will allow water to run down the lines and drip beneath. Further, there should be attachment points in all four corners as well as at least one point in the middle of each edge.
I'd suggest going on YouTube and looking at some of the tarp videos. Many are quite thorough, including the various knots you will need to learn and methods of rigging and positioning the tarp for various use cases. I learned a lot when I was first shopping for tarps.
Finally, I would not buy a tarp online without having seen it in person. It's difficult to judge the quality of construction or durability of the material without actually having it in your hand.