Are the DNA codon mappings arbitrary?
There is no reason for why a certain codon came to represent a certain amino acid. But some reason is attributed to similarity of degenerate codons. Usually the degenerate codons only differ in their last (third) nucleotide. The corresponding first anticodon in tRNA usually bears a modified nucleotide such as inosine which can (promiscuously) pair with A, C and U. 

You can see the wikipedia page on wobble base-pairing.