It is called a proliferation and can be potted up and you will get an exact copy of the mother plant!!!
How I do it is to cut it about an inch under the prolif and an inch above and then stick the bottom into soil just until the bottom of the proliferation touches the soil. The inch of stem helps hold it until it roots. Cool way your garden gives you free plants. I only got 3 this year but consider that good since the daylily year was sort of blah in the heat.