1/4" to 3/8" or whatever, the only rule is the height of the string as it rest on your nut of choice in relationship to your first fret, you want approx .030 clearance to your first fret, so it's all about your build, larger nut may require more recess cut/relief into the fret board, I actually used a 3/8" on my first build but the nut sat just at the end of the fret board, not on it, and started my fret measurements from there when I cut the rest. My current build won't have a nut, but a zero fret, I'm using a jumbo fret for my zero and medium frets for the rest, adding a neck back angle helps with this and gives you super low action. I'll have a lot of normal fiddling with all the frets to make sure they're level to each other to avoid buzzing, but should be worth it when I'm done.
For a slide git, any threaded thingy you have on hand that gets the strings off the neck.
For a fretted git, don't do it. it has a different 'last contact' point than a regular nut and will throw off your fret positions & intonation if you don't know where to compensate from, and its much more difficult to file the nut slot in the neck to get a bolt down lower for the string action height, compared to just filing string slots in a wood/antler/bone/corian nut. and if you mess up, and go too low? well a nut is easier to toss and re-cut than a neck...
don't do it..? tons of people, thousands have used bolts for nuts, practically part of the culture anymore to come up up with whats on hand for any component of these gits, and filing the slot is no more difficult the the rest of the build. If he wants a traditional nut, he wouldn't have asked what size bolt would be best, to answer that better, maybe a 1/4" x 1-1/2" long.
No offense JL, not a personal attack on you, a traditional bone or plastic nut would be easier per se, and he can go that way, but to say don't do it is telling everyone to...well, I won't beat a dead horse.
And my apologies in advance if I didn't convey properly this isn't a personal attack on your opinion.
I use a bolt as a nut for most of my fretted guitars. As long as you cut the slot for the bolt centred on the "zero fret" position the intonation should be bang-on right. Once you get the hang of how deep to set the bolt, I find it is much easier and quicker than installing a conventional bone or plastic nut. I use 5mm, which is a bit smaller than 1/4".
nice design, I like it