Rooting Hibiscus cuttings is actually pretty easy. Take 4 - 6 inch cuttings from softwood non-blooming tips. Cut all the leaves except the top two and cut those in half. Cut the stem end just below a leaf set (node) with a sharp knife (pruners damage cells).
Method 1: Put the cutting in a jar of water with a drop of added hydrogen peroxide. Keep the water clean but everytime you change the water, add a drop of peroxide.
Method 2: Plant the cutting in potting a soil/perlite mixture (half and half). Make sure the soil is thoroughly wet. Dip the end of the cutting in hormone rooting compound. Poke a hole in the soil with a pencil or your finger, insert the cutting into the soil to just below the leaf set and firm the soil around it. Build a tent with a plastic bag (cut one corner off so the humidity doesn't get too high) and wait. Keep the soil damp.
Tropical hibiscus don't root as quickly or as well at Rose of Sharon so take several cuttings. Plant them all in one pot - the pot should be big enough so each cutting has about an inch of space around it. Cuttings planted together have a better chance of success than cuttings planted alone. Maybe they get lonely?
With either method, you should have roots in a couple months. Water-rooted cuttings are much more fragile than soil rooted cuttings. Water roots are very fragile and easily broken during the transplant to soil.