I like the one motor / multi drive idea but if your wrapping line don't all the hooks have to spin the same direction? If so the over/under idea wouldn't work. We had a machine to twist fibers into rope in Boy Scouts and I think thats how it worked, it was only a 3 or 4 strand setup though that you feed that cheap single strand sisal twine for nursury use into to make a heavier rope. Gears would be ideal but that moves it away from the build it yourself realm.
If you could one large stationary gear with say 200 teeth, your 10 hooks mounted on a rotating outer ring driven by your motor with say 10 teeth running on that stationary gear, that would twist the induvidual lines at a 20 to 1 ratio and follow the twist of the finished line as it was being layed down. Or would you need an outer stationary gear toothed on the inside for proper rotation? I'd have to twist a few strings to get the two rotations needed.
You would need a good machinist friend though.
If your not twisting too tight a full circle belt around 10 pulleys might work, a toothed belt and pulleys would definately work.
Rubber wheels on the ten shafts running againt a stationary hub as the geared example above might work also if your torque requirments are not too high, that's in the realm of do it yourself.
sorry rambling out loud, been in the drivers seat for 27 straight hours