That's an excellent answer. In addition, two other things to consider:

1. the use of the number three continues to this day, due to the Christian ideals that have later sprung from Greek (and then Roman) societies. You can see examples in everything from religion (the Holy Trinity) to literature (Dante's Divine Comedy) to movies (how many trilogies have been released in the last 10 years, from the Matrix to the Bournes series to Star Wars to etc, etc, etc)

2. Repetition occurs in numbers other than three, and its important to keep in mind that as the Iliad and the Odyssey are originally oral stories, repetition (like the use in the Odyssey of Dawn's "rosy fingers" in many chapters, at least of the Fagles translation) helped save memory for other things. Basically, it was one less phrase that had to be memorized later.

Dave Becker

The number three is a recurrent theme in Greek mythology. For the Greeks, this number was understood to mean several things.  First, "three" could stand for the three most powerful gods, Zeus, Poseidon, and Hades, and their respective realms: Heaven, Earth, and the Underworld.  Three also stood symbolically for the three Greek Fates (Clotho, Lachesis, and Atropos.)  Three is also the number of prongs on Poseidon's fork, and three was indicitive of the different aspects of death (Thantos, Ker, and Gorgo).  Hecate is also the three-faced goddess.  Depending on the way three is used in the Iliad, it could mean one or more of these things.