Why are there three stop codons but only one start codon?
Disclaimer

It is important to realize that it is not possible to answer questions of this sort (the evolution of translation)  in anything other than a speculative manner. Please bear this in mind, especially if you find the speculation below persuasive.

Summary

It is suggested that the redundancy of termination codons is largely similar to that for ‘elongation codons’ decoded at the A-site of the ribosome, although protection against suppressor tRNAs may also be a factor. The consideration here is, therefore, mainly focussed on the question why there is only a single codon for methionine (initiation and elongation), and I suggest this may have been a combination of the late appearance and infrequent usage of methionine, together with the particular requirements for an initiation tRNA at the P-site of the ribosome, where standard wobble is apparently not possible.

Disposing of a Red Herring

Although alternative initiation codons are sometimes used, this usage differs in prokaryotes and eukaryotes, is very limited, and is not seen with the use of methionine in elongation. I would therefore consider this does not invalidate the question, but perhaps rephrase it as: 


  “Why does the genetic code have only one codon assigned to methionine,
  the initiation codon, whereas other amino acids and termination have
  generally more than one?”


Time of appearance of and extent of use of Methionine

It is generally thought that genetic code initially encoded a small number of amino acids, and that this number increased over time. Methionine is one of the most complex (and energetically expensive) to synthesize (see Wikipedia entry), so is likely to have appeared late. One can argue that most of the codons would have been ‘taken’ (assigned) by that time. This argument is consistent with the situation for tryptophan, the other amino acid which has only a single codon, and which is also synthetically complex. Furthermore, these are the two least abundant amino acids so on can argue that their number of codons reflects their relative usage.

Admittedly, there need only be one stop codon per protein, but it is possible that the mechanism of termination evolved before that of methionine-specific initiation. (Initiation might originally have been from the 5′-end of the mRNA.) There is also the question of protection against suppressor tRNAs, but I shall delay discussion of this until the end.

The unique features of the initiator tRNA

Elongation involves several dozen amino acyl-tRNAs being recognized by an elongation factor which binds them to the A-site of the ribosome, at which 3′-codon wobble occurs where the redundancy of the genetic code allows and the anticodon of the tRNA has evolved appropriately. 

In contrast initiation involves exclusive recognition of a special tRNA by initiation factors but not elongation factors, and subsequent binding at the P-site The tRNA has unique structural features that allows it to be N-formylated — structural features that are retained in eukaryotic initiator tRNA, even though there is no transformylase in eukaryotes. (see this SE Question]

This uniqueness may explain why there is only one tRNA for initiation. However one might ask why there is a not a group of methionine codons (e.g. AUA as well as AUG) recognized by the same initiator tRNA through wobble. The reason for this would seem to be that the P-site does not allow third position wobble (as I explain in the answer to a related question), so the initiation met-tRNA would be codon-specific.

Loose Ends

Although perhaps straying a little from the question, one might still ask why there are not alternative elongation codons for Met — e.g. AUA, which could have been ‘stolen’ from Ile, which already has AUU and AUC. In fact in mammalian mitochondria this codon reassignment has happened, but unfortunately does not help explain things. In mitochondria there is only one tRNAmet for both these codons, consistent with the simplification of tRNAs for the small mammalian mitochondrial genome. Initiation requires the formylation of the methionine.

I will finish by returning to the question of multiple termination codons, which are not really consistent with my suggestion that the extent of redundancy reflects usage. Although termination codons, like elongation codons, are recognized at the A-site they are recognized by protein release factors rather than tRNAs. This means that they will be affected differently by a mutation in a tRNA that alters the anticodon. For an amino acid codon, an altered non-cognate tRNA would only cause some misreading and an altered cognate tRNA might be neutralized by wobble or by duplicate or redundant tRNA genes. AS termination codons are not normally decoded by tRNAs, a suppressor tRNA would cause read-through at all the suppressed codons. The presence of two alternatives means that termination would be likely to occur within about 30 codons 3′ to the original termination codon. Variation in the length of the N- and C-termini of proteins is common among related species, and thus need not be deleterious to the cell.

Related SE Questions 


‘Wobble’ in Initiation and Elongation — I have contributed an answer to this.
Why is AUG the Initiation Codon? — I will respond to this presently.
Redundancy of the Genetic Code — I have contributed an answer to this.
Initiator and Elongator tRNAs in Translation — I have contributed an answer to this.