Sheet pants! During the summer when my girls were young we would take a pants pattern, and an old soft sheet and make elastic waist pants out of the sheets.
My daughter learned to sew with an old sheet. She made a doll and a little bed for it, all on her own with my sewing machine and scraps and an old sheet.
The unworn edges of the sheet that get tucked in can be cut into squares for pot holders. Add a piece of old blanket or quilt batting between two squares, sew together, turn and stitch along the edges and its a pot holder.
Old sheets are ideal for applying stain. If the sheets are in good condition you can make pillow cases, or double up the fabric and make throw pillows depending on how worn the sheets are.
They also make great tote bags if a person enjoys sewing.
A small piece of a sheet added as an embellishment for scrapbooking is good as well. A picture of your child, and her room from days gone by with a tiny piece of the sheets she used to have on her bed.
You got me started, now I want to write a hub! :-)