Here is the latest I have found. link
This is the basic idea: Evolution does not look ahead and make plans. It would not create a system of mRNA with giant ribosomes to create proteins, until proteins were already useful. There must have been a simpler way to create useful proteins before, that got complexified after it was already useful.
Since enzymes can be made of RNA, presumably early life made just RNA enzymes and no proteins. It didn't matter whether RNA enzymes were as efficient as modern ones because protein competition hadn't evolved yet.
And since we now make peptides off an RNA template, probably early proteins were also made off an RNA template. It is hypothesized that particular amino acids would be attracted to particular RNA pairs, and they would be put in just the position that would help them link to neighboring amino acids. This could be slow and inefficient compared to modern protein production because again the competition was not stiff back then.
Between two pairs of coding bases, the RNA needed a noncoding base for spacing. Possibly the noncoding base might be particularly good for catalyzing the joining of amino acids.
The system did not have to use the same amino acids we use today. It could use amino acids that were good at attaching to RNA, and later after the system got changed around the modern list of 22 amino acids could have developed.
There are various ways this is a satisfying idea. For example, the machinery to make proteins is largely made of RNA. The mRNA carries the message. tRNAs decode the message. Ribosomes are giant machines which transcribe proteins efficiently, and they are largely made of RNA with a lot of little proteins that help them hold their shape etc. Presumably the first protein-building machinery was itself built entirely from RNA -- because that's what was there to build with -- and over time evolution found ways to improve that machine with protein scaffolding.
This 2005 paper link describes a reasonably plausible way it could have been once. They imagine a few alternate amino acids which might have once been part of the genetic code -- ornithine. Homoserine. 2,4-diaminobutyrate. They imagine ways that the crevices between pairs of bases could be like the binding sites of enzymes. They fit the ideas together.
Have there been advances since 2005?