Choose Your Material
Fleece (sweatshirt fleece, polar fleece, etc.) is my favorite fabric for making homemade hammocks, You can also experiment with different materials for different purposes: fleece and flannel make cozy beds for winter time, but cotton and denim make nice beds for warmer weather. More on fabric choices and finding inexpensive sources can be found in Choosing Fabrics for Homemade Hammocks (no matter what fabric is used, watch for loose threads and discard the hammock if it is being chewed up).
Clips for Hanging the Hammock
I've had to discard a few store bought hammocks due to chewing, but I saved the swivel clips of these for making my own hammocks. You can also by a variety of clips, snap hooks, carabiners, shower curtain hooks and other assorted hardware or jewelry making supplies that can be used to attach hammocks to cages. Check your local craft, hardware or animal supply store for different options. Whichever option you choose, try to think about it in terms of safety, making sure no toenails, feet, heads or other body parts could get trapped in the clips you choose.
The Hanging Loops on the Hammock
Bits of fabric, sturdy ribbons, bias tape or nylon webbing (like leash material) can be used for making the loops to which the hanging clips are attached. Thin leashes bought at discount stores can be cut into smaller sections as inexpensive webbing for the hanging loops.