Cotton Chiffon Fabric
4. 100% Cotton Batiste or Broadcloth: you can find beautiful pure cotton batistes and broadcloths usually intended for heirloom sewing. They are a bit pricier than blends but make the most sense if you prefer entirely natural fibers.
5. Radiance Cotton/Silk Blends: Another notch up on the fancy scale! This is a cotton and silk blend fabric that is often sold in quilting shops. It has a slippery texture on one side and is opaque. I've lined all sorts of garments with it (including wool skirts) and loved the results. Better for medium weight garments. Fabric.com carries a good selection for $16 yard. Because of the price, I reserve this for special garments.
6. Stretch Cottons: If you use a stretch fashion fabric, you'll want your lining to stretch as well. Stretch linings are hard to find and are usually polyester only, so I recommend looking in the fashion fabric sections. Anything called "stretch poplin" or "stretch shirting" should be a good match. Fabric.com has a bunch of cotton blend stretch poplins for around $6/yard, but I've never used them. Let me know if you have a preferred brand/source!
Keep in mind that you don't have to use the same lining fabric in the bodice and skirt of a dress. For my blue and white brocade dress, I used cotton/silk in the bodice (for comfort), and silk crepe de chine in the skirt (for drape and slinkiness).
You can choose a more expensive, natural fiber fabric for the bodice and then use a cheaper synthetic for the skirt since it won't be as close to your skin. Or use a medium weight lining for the bodice to provide structure and a lightweight lining for the skirt for easier movement and flow. So many options.
I hope this helps a bit. Please share your favorite summer linings too!
P.S. A quick shop update (ignore if you're on one of those stash-busting fabric diets I keep hearing about-can't say I've ever tried one of those!). I added a bunch of new stuff recently (and will have more next week).