Why did translation develop a specific codon for initiation?
It is not clear to me where initiation took place in the translation of artificial polynucleotides such as (UC)n which Khorana used in experiments that, following up the work of Nirenberg and Matthaii, helped decipher the genetic code. It may well have occurred at random positions rather than the 5′ end. Such a mode of translation would obviously have been wasteful if it occurred during early evolution as a mixture of products would be formed, many of which would be inactive and might even compete with the active species. If this were the original situation,  it is easy to see how a change to initiation at a specific point would be advantageous. 

But why not start at the 5′ end? Contemporary eukaryotic initiation complexes recognize the 5′ end of the mRNA, even though they then scan further to find a specific initiation codon that will respond to the initiator tRNA. As there is no physical separation of  genes on a chromosome, it is obvious why transcription needs to start at a specific point within the DNA, but the need not be true for the translation of individual mRNA transcripts.

I can see two ways of approaching this question. 

The first is from the standpoint of the contemporary ‘DNA World’. The arguments one can make  include:


In a general sense 5′-end initiation may be regarded as restrictive
in that it would not allow the 5’ end of mRNA to provide other
functions such as forming a secondary structure to regulate the ease
of translation.
5′-end initiation would preclude the polycistronic mRNAs we see in
contemporary bacteria.
In 5′-end initiation it might have been difficult to distinguish the
ends of mRNAs from other RNAs before the modification of bases that
we see in contemporary eukaryotic mRNAs had developed.
Likewise, as suggested by @John in a comment on the question, 5′-end initiation might have caused translation of fragments of RNA as they degraded.
The selection of a specific codon was secondary to the need for a
single initiator tRNA, which has been discussed in relation to
another SE question, although the arguments there relate to more recent stages in evolution.


However there is another approach which considers the situation in a putative ‘RNA World’.

The difference here is that — like some contemporary RNA viruses and bacteriophages — the genome might also have been the mRNA but without a system to produce many copies of the ‘plus’ strand for translation. Translation may have been on the genomic RNA. 


To translate several proteins it would have been advantageous to have
several initiation sites on what would have been a polycistronic
messenger (rather synthesize a polyprotein like poliovirus).
The 5′-end of the RNA may not, in any case, have been available for
encoding protein as it might have another (e.g. replicative)
function, and it is also possible that part of what in contemporary
systems is a separate part of the translation apparatus was
physically incorporated into the genome.