Aluminum or PVC frames on lawn chairs last longer than the plastic strapping they come with. You can replace worn and broken strapping with more plastic strapping or you can get creative.
Cotton duck or sail cloth is sturdy enough to be used for lawn chair seats. A heavy duty thread and upholstery needle is used to attach the material to the frame by hand-sewing. Also simple to do, with the proper tools, is creating grommets in the fabric and lacing the material to the frame or using a snap tool to fasten the material directly.
Poly rope is similar to the cotton rope seen used for hammocks but is not a natural fiber. This makes it a bit stiffer to work with than natural fiber ropes or cords, but overall is a good choice. Its longevity surpasses cotton and other natural fibers and it doesn't require sealing to make it weatherproof.
For a retro look, try replacing the strapping with webbing or flat strapping. Webbing is attached by screws to existing aluminum frames. Repair kits can be purchased in hardware stores. Flat strapping can be used to repair the tubular plastic strapping found on most department store lawn furniture. Video tutorials are available and many pictures and testimonials of successful make-overs can help you decide how to give new life to your old patio furniture.