Rather than clipping gear randomly onto your harness when cleaning, have a dedicated system and stick with it. This one works great.
Before the second leaves the belay, they drape two single runners over their shoulders, one on each side. When they clean, all of the protection (cams, stoppers, etc) gets clipped to one side, say the right, and all the quickdraws, runners and extra carabiners get clipped to the left. Then at the next belay, the second can easily lift these slings off of their head and quickly rerack the gear.
If they keep this pretty organized when they are cleaning, at the belay all they have to do is grab any remaining gear from the leader, and then the second is ready to head out on the next pitch.
Here’s a short video clip by big wall expert and Supertopo founder Chris McNamara taking about this cleaning technique (starts at 1:25).