If it's really digging into your neck, maybe check how your guitar is slung and try and find a better position for it. Or try a different strap - it may have a sharp side on it. Thick leather straps can have sharpish edges, and nylon straps can dig in if they don't go over a bit of you covered by a T-shirt (not recommended on sunburn! :-).

A shoelace is ideal for tying to the nut. You can tie it right at the nut (underneath the strings) but I find this gets in the way of my fingers - on my classical, I prefer to lace it around the centre portion of the head which keeps it well away from my left hand. Best bet wherever you put it is to take the string round the guitar neck twice (or more) so that there isn't any tendency for it to slip around. And remember to tie the lace in a bow, otherwise you'll never untie it again!