Honeysuckle seems to grow abundantly in my neighborhood in Virginia. I have read online about honeysuckle being an invasive species many sites refer to it as a "bad guy".
I have considered growing it in the backyard to attract birds and for the fragrance.
Is there anything inherently bad if I were to grow honeysuckle in my backyard from a clipping?
Just be sure that you plant the right type of honeysuckle!
That white/cream colored honeysuckle that you are seeing/smelling everywhere right now is Japanese honeysuckle. It is extremely invasive, and once it is established, is nearly impossible to eradicate. I also live in VA, and have personally seen older roots as big as my wrist.
You mentioned birds. Coral honeysuckle vine is a variety that is native to VA. It is good for attracting hummingbirds, and is grown at the Norfolk Botanical Gardens. http://www.miraclegro.com/smg/goART2/InfoHowTo/grow-coral-honeysuckle-vine-for-hummingbirds/16400044
(Jim Young in comments had an answer, too.)
Why not grow a honeysuckle cousin, Honeyberry (Lonicera caerulea)? With this non-climbing fruiting bush you get flowers in spring followed by tasty edible berries (similar to blueberries).
Honeyberries are native to eastern Russia and hardy to -40F (zone 3), but can be grown as far south as USDA hardiness zone 8, so in Virginia I see you having no problem growing it.
