Where should the dots be on a guitar fretboard?

I believe that you should have a dot on the 10th fret, and not at the 9th. Let's focus on this specific dot first, and then talk about the rest of the dots.

1. They are reference points. You learn the notes relative to them. For this, it doesn't matter where they are, as long as they are not too far apart. So this does not suggest position 9 especially.
2. They have musical meaning, such as 4ths/5ths/octaves, or represent the position of harmonics. The first justification (intervals) is weak. If you are playing in a random key, and working from a random note, those interval markers are unlikely to be what you need. There is thus no compelling reason for a dot giving the 6th degree, or the tonic of the relative minor, of an open string, which would be justifying it after the fact anyway.

   There is a reasonable justification for the octave. Those reference points for the octave of 6 open-string notes (despite the fact that they may be rather irrelevant to the current key) helps you negotiate around. The octave is such a fundamental concept in all our minds. The one about harmonics is very weak. You don't often use harmonics, and anyway, you will easily remember them after a short time, especially if there are some dots somewhere.

3. They are of practical use in helping the player with the specific characteristics of the instrument, rather than any musical theory use.

For my money, it's reason 3 that is compelling, and makes a great case for the 10th fret instead of the 9th. That first dot at the 5th fret helps you to shift strings. If you are plucking away and it's inconvenient to play the E3 on the D string, then that dot at fret 5 tells you that the A string can act for the D string from the dot on up. So you pluck your A string two frets up from the dot, instead of the D string 2 frets up from the nut.

Now, notice that the dot on the 10th fret is the same interval again up from the 5th fret. So in the above example you can get your E3 by moving from the D string down two strings, and playing the low E string two frets up from the dot at the 10th fret.

A very compelling example: Say you wanted an F#3 as on that D string. The eye/brain knows that it's one fret down from the dot at the 5th fret. Easy to switch to the A string, one fret down from the dot on the 10th fret.

Miracle of miracles, the standard dot at the third fret above the octave, is at fret 15! The same as the typical string-change interval, once again.

Thus, ignoring the annoying anomaly between the G string and the B string, those dots at frets 5, (ideally) 10, and 15 help you cross from string to string.

I actually use all-4ths tuning: E-A-D-G-C-F and of thus course the dot at the 10th fret is of even more use. I happily play all over the fingerboard, with those string-crossing signposts to help me.

To "move" the dot, I first used a black or brown permanent marker to cover the pearl dot and the little dot on the side of the neck at fret 9. At fret 10, I put a dot on the fingerboard and on the side of the neck, using white nail polish. To paint that new fingerboard dot, I cut a little hole in a stick-on label and used it as a temporary mask. One day I might get a luthier to make a better one.

Actually the "permanent" marker wore off, exposing the original dots at fret 9. That's when I applied black nail polish onto the fretboard and the side of the neck... with success.

So as for the rest of the dots, my final dot layout is: 5, 10, 12, 15, 20. I used the nail polish to remove and add dots to achieve that. The only interruption to this logical regular layout is the octave (double) dot. I really don't need it, and it probably distracts from the logic, but I partially painted over it and left two small dots in honor of the factor of two. My final comment is that the brain makes negotiation of the fingerboard automatic, after enough practice -- with any dot layout or no dots at all -- but this dot layout is a great help, and enabled me to learn the whole fretboard quickly, after a bit of sightreading practice.