This question is mainly for my banjo but it seems to apply to my guitar as well. For me it is most comfortable to have the left end of the strap attached to just above the nut on the neck. Is this a bad idea? I worry that it is putting stress on the neck.
In the case of a heavy banjo (and specially if it has a long neck) I wouldn't recomend that you attach your strap that way, it would most surely cause tuning problems, to say the least. For a guitar I don't think it would be such a big deal. Been using mine that way for decades, without any problem.
In general, guitars are made to balance with the strap attached to the body (I know, it's a generalisation - one of my acoustics has the strap attached at the nut:-) - so altering that will change the way it hangs. Whether or not that is a bad thing is obviously up to you.
I would imagine, even for an electric, that it shouldn't harm the guitar - the truss rod and neck are set up to cope with high levels of tension so unless you jump up and down a lot, the bending moment on the neck shouldn't be an issue.
Since the weight of a banjo is primarily in the pot, we players tend to have the straps anchor the pot itself. We never put a strap on the neck near the nut.
It's usually configured like this: http://neotechstraps.com/skin/frontend/neotech/default/images/Pilot/SuperBanjoStrap2.jpg ...and you choose the hooks to attach to based on the ones that leave your neck at a natural angle when the pot is hanging in front of you. For example, if the neck naturally hangs low, snug the left strap farther along on the circumference of the pot hooks. Basically: Balance the circle (pot) so that the neck naturally hangs where you need it.
Our necks are much thinner than a guitar neck, so we A) Don't want to expose it to stress, and B) it's light enough to be balanced by the pot, itself.
And as a reality reference: My right strap is one hook "above" the tailpiece anchor, and my left strap snugs around the backside/heel of the neck and grabs the second (from the heel) hook from there. So to say it another way, if you're looking straight-on at my banjo with the neck up at noon, My right strap is around 6:15 (hourhand-wise), and the left is about 1 o'clock.
Necks tend to be thin and primarily designed to deal with the forces of the strings. Hanging the whole instrument from them adds stress quite in a different direction and outside of its design.
The heavier the instrument and the slimmer (in the direction across the strings) and longer the neck, the more effect the "unexpected" forces can deal out. For "sideways warp" there are no correction mechanisms.
Does it matter with your particular instrument and playing style? In the end, that's for you to decide.
For a very lightweight banjo with a thick neck, you can indeed attach the strap to the neck or headstock. The banjos made by Backyard Music weigh a bit less than two pounds, most of it in the neck, and have a relatively thick neck. These banjos come with a screw-eye in the headstock for attaching a strap:
You can also see the relatively thick neck that, along with the light weight, makes attaching the strap to the headstock not be a problem.
I have one of these banjos, which is how I know this. Mine has had no ill effects from having its strap attached to the headstock: It hangs comfortably and doesn't go out of tune whether I wear the strap or not.