With only 2 nucleotides per codon, a codon could only encode 16 amino acids (actually just 15, because you need at least one stop codon), which isn't enough to encode the 20 acids we require (actually 21). So 3 is the bare minimum. (Of course another option would have been to introduce two new nucleotides, but evidence suggests that just using 3 nucleotides per codon was easier)
Nature has a tendency to minimize everything (except entropy :D). And apparently evolution selected those who understand that:
Simplicity is the ultimate sophistication.
- Leonardo da Vinci
Perfection is finally attained not when there is no longer anything to add, but when there is no longer anything to take away.
- Antoine de Saint Exupéry
It's rather safe to assume that the 20 acids we use are the bare minimum we require to function and that's why we don't have more.
Ultimately, RNA/DNA is there to store amino acid sequences. So the number of nucleotides in a codon is determined by the number of acids that need to be encoded and not the other way round.
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