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Herman:

When you were feeling overwhelmed by having "too many" lines of position, I feel like the only conflict you should have been having is between "closing your eyes and pointing" and "arbitrarily choosing round or otherwise convenient numbers."  It might be helpful to consider that six LoP's is better than three LoP's, but not twice as much better.  Here in the eastern US, on land, three lines from a handheld sextant can generally determine which county I'm in (10's of miles across); more lines can help confirm my county, increasing your confidence, but (all things being equal) I still won't be able to establish my town or postcode (1's of miles across or smaller) with any reasonable confidence, with six lines or sixteen.

As for your three-by-five puzzle, if you're relying exclusively on tables, you're probably better off using 3 bodies, 5 sights of each, as it's easier to compute the average of multiple sights of a single body than to compute a single sight reduction for a particular body.  But if CPU cycles are free, the right answer is definitely to use 5 bodies, taking 3 sights of each.

Taking multiple sights of the same body and averaging them together helps to reduce the probability of random errors in measuring that body's altitude, but that random error can never be eliminated and there are diminishing returns when making multiple sights to average.  A second sight may increase your certainty by, say, an additional 25%, but the third will only increase it an additional 12.5%, the fourth an additional 6.25%, and the fifth is likely not worth the effort.

However, truly random errors are not the only kind of error to worry about; there are also constant errors.  If you're dealing with, for example, an unknown index error or unknown dip/refraction error of +0.2', all of your readings are going to be 0.2' too high, and the average will be 0.2' too high no matter how many sights of a single body you average together.  The only way you'll compensate for that is if you measure the altitudes of different bodies in different directions and see how the different lines of position relate to each other.  There are diminishing returns here as well, but taking a sight of an additional body increases certainty in two dimensions (lat & lon of your fix) rather than just one (towards versus away).

When all is said and done, though, 15 sights in a single session may be too many to be useful anyway.  Fatigue introduces its own random error, one that increases in magnitude over time.  I feel having access to computers should encourage you to take sights of more bodies in a single session, but perhaps not necessarily more sights (i.e. more bouts of  "twiddle the knob, rock the sextant, squint at the vernier") per session.