lpf3 pretty much nailed the high points. Like he said, Crazy Glue, the gel type, is pretty good for a quick fill, especially where you're working with a clear only finish. It will be noticeable, and it isn't very durable as a finish, but it can be used as a quick partial fill. Keep in mind that if you do use the Crazy Glue it will make the repair more apparent.
If you want to use a hard finish to repair the nick then you need to determine what that finish is. My guess is that you've got some sort of poly or catalyst finish on the neck. In which case lacquer might make a nick a truly ugly spot. If you want to make sure just pull the neck and try a spot of lacquer on the back of the heel. If it lifts it then who's to notice...and you'll know it's not a lacquer finish on the neck...
If it just melts in and only makes a high spot then it's lacquer on the neck...but I'd bet what little hair I have left that it ain't... I mention this as lpf3 alluded to a tint in the finish and how hard it is to match it. It's actually pretty easy with lacquer. All you do is take some clear lacquer, put it in a clear glass jar and let it sit in the sunlight for a while. Soon enough you will have a pretty good match for your tinted lacquer repair. ...just a little bit as a piece of trivia...
What I do on nicks on the neck is actually pretty common and straight forward. If it's down to the wood then the odds are you've stained the wood with the oil from your thumb. Easy enough to remove. Take a Q-Tip, some 90% rubbing alcohol, dip the Q-Tip in said rubbing alcohol and lightly dab it on the bare wood at the bottom of the nick. Wipe the alcohol away with the dry end. Repeat as often as necessary to get back to a clean surface with no discernible crud in the nick. I let this sit overnight, but a couple of hours are fine.
When you do get back to it get some 400 grit wood sand paper, not the wet or dry, but this stuff:
...or a 3M pad, like these:
You don't want to sand the edge of the nick away, as that's what's going to hold in the repair finish. Just gently knock off the gloss around the nick and make sure the wood itself is smooth.
Take a tack rag after this and wipe the area down so no sawdust or debris is in the nick.
Now for the finish. If it were me, I'd get some spar urethane in the final sheen to match the existing finish. (IE: High Gloss, Satin...etc...) A very small can is all you'll need...but try and find one of those. Stay away from a spray. No one's aim is that good...
Now grab a small artist brush...maybe two or three...like these:
I mentioned getting several, because cleaning poly off a brush is a pain. Acetone will do it...so will the really cheap finger nail polish remover...which is 99% acetone...but if you don't get all of it off the brush is fodder anyway... You could just wrap it in a plastic bag, but honestly, you're going to have the guitar longer then the brush, and you want the guitar to look better then the brush...so buy 4-5 and just toss them after each coat.
What you want to do with poly is to build it up in light coats. This is not a 1 hour fix. Just put enough on you brush to coat the inside of the nick. Let this dry the recommended time between coats, go back over it with your 3M pad or 400 grit sand paper and repeat as ofter as necessary to fill the hole. Remember to hit your nick with the tack rag before you put on the next coat. Poly will not melt the leavings as well as lacquer, so make sure it's clean before you go to the next coat.
I go a little over the existing finish and sand it back down until it's flush.
I should mention how I sand small areas, as well as in between these interim coats. I generally find a small piece of wood, maybe 1/4" wide and sand it smooth and wrap the sandpaper around it. This means you'll wind up with dozens of small pieces of sandpaper laying around, but this method keeps you from sanding to hard or too far a field from the nick.
Once I get the nick repair flush with the existing finish, then I break out the wet or dry and paper. I use 400 grit and 800 grit, then 0000 3M pads. Here you can use your fingers. As you go up in grit make sure all the scratches and lines are gone BEFORE you go to the next higher grit. Once you have the nic smooth, then forget about it for about 2-4 weeks. Let the poly cure.
You can still play it, but just live with the dull spot until the poly completely cures. This varies from manufacturer to manufacturer, but it generally falls into the 2-4 week category.
Once you've waited your obligatory time, then you can buff it out. Yes, Virginia, you can buff poly just like lacquer. Start out with some polishing compound and lightly work on the dull spot. Then move up to a swirl remover. I use this stuff:
...and after that, if you still have any feeling left in your fingers, finish it up with some of this stuff:
It's a slow process when doing it by hand, but when you're done it'll look just like the new and only you will know where it was.
Or, you can just buy a new guitar...
Happy Trails
Cynical One
to the NutzHouse!
) and after photos to add to this thread. 
