let it root on the plant before taking the cutting. Scarify an area apply rooting hormone and put some soil mix around it use saran wrap or something to hold it in place, keep it moist and on 2 to 3 weeks you will see roots. Then cut below the roots and you have another plant!
You can also do it an easier way, if the honeysuckle is long enough to touch the ground, spread it along the soil and let it root. Once it roots dig up the rooted areas and clip it off the main vine.
I have that stuff everywhere too. I know it'll root like that because I'm constantly pulling it up---- argh.