Snip off a 2- to 3-inch shoot of your coleus, using sterilized scissors or gardening clippers to snip off a 2- to 3-inch shoot. Choose one that is thick with leaves and in the middle of the plant. Cut it directly above a leaf node, which is the point at which leaves emerge from the shoot.
Pinch off most of the leaves from the shoot, leaving only a couple of pairs of leaves at the tip.
Fill a shallow glass container with water and let it sit until it reaches room temperature. Position the coleus shoot in the water so that the cut end is submerged, but the leaves are dry and above the edge of the container.
Place the container in a window that is exposed to indirect sunlight. Add more water when the water level gets low, and replace it with new water if it becomes murky.
Watch the cutting carefully. It will quickly grow roots. After about two weeks, the roots should be several inches long -- long enough to plant in soil, according to This Old House. Plant the cutting in rich potting soil and watch it quickly grow into a new, colorful coleus plant.