Honeysuckle will not root readily in water. There are a couple of ways that you can propagate it, however. The simplest is to layer it. Take a branch or stem from near the ground and bend it over, burying and pinning down a section of the stem. It helps to make a small wound in the buried section and to apply rooting hormone to the wound. It will take a full growing season for the stem to develop new roots.
The other method is to take hardwood cuttings after the plant has gone dormant in fall. Use stem tip cuttings about 6 inches long. Place the cuttings in rooting media such as perlite, vermiculite or peat/perlite mix. Dip the cut end of the stem into rooting hormone before sticking it into the rooting medium. Maintain high humidity and bottom heat to induce rooting. Ideally, you'd have a mist bed that showers the cuttings to keep the humidity hear 100 percent. You can also cover the cuttings with plastic film to maintain high humidity. It may take several weeks to several months for new roots to form.