I start with the supposition that each nucleotides has the same amount of information,
How much?
and also that none of them is statistically favored to be a beneficial mutation. Therefore, natural selection, which favors beneficial mutations, has an equal amount of chance to favor one nucleotides or another,
None of them being statistically favored doesn't necessitate that NS has an equal chance to favor one nucleotides or another. Beneficial mutations are going to be made of different nucleotide sequences which is where we get the non-random "connection". We can't assume that the nucleotide sequences, themselves, are not going to have an affect on how beneficial a mutation may be.
Of course, if we were to discover that natural selection favors some nucleotides over others it would probably also merit a Nobel prize ...
It seems obvious that it does. A prize should be offer to the ones who find out how that 'connection' can be predicted.