After considering this question some more, I think a key feature for this hypothesis to work is that the original function must be so integrated and vital for the biology of the organism that it must be mimicked/echoed to some extent. In terms of your RNA-example, it was not enought to evolve the ability to synthesize any complex molecule that could function as an enzyme and information carrier, but had to be RNA since this was what the biological pathways of the primitive organism was already using.

With this in mind I think that you should look closely at vitamins (or other essential nutrients) for evidence of similar processes. The key think about vitamins is that they are often large complex molecules that are vital nutrients and the ability to synthesize vitamins differs widely between taxa. However, keep in mind that molecular biology is not at all my field, and I might be mistaken on some aspects.

Take the example of Vitamin C. This cannot be synthezised by humans, but by most other mammals (vitamin C synthesis is considered ancestral in mammals). It can also be synthesized by plants, yeast and many other animals. However, vitamin C is used in different ways in different organisms. Plants use it e.g. for photosynthesis, photoprotection, cell wall growth and in the synthesis of plant hormones, while humans and other mammals use it as enzyme cofactors and for immune responses. Another key aspect is that plants, yeast and animals all use different pathways to synthezise vitamin C (Drouin et al, 2011), which indicates that the synthesis of vitamin C is not ancestral to all these taxa (i.e. not an ancestral feature of all eukaryotic life). In mammals, the synthesis of vitamin C has also switched between organs multiple times, which also indicates novel adaptations that results in the same molecule.

To me, it seems very likely that vitamin C was initially obtained by animals from plants (which have very high concentrations of vitamin C) as part of their diet (i.e. environmentally provided) and became an essential component in many biological pathways. Later, animals gained the ability to synthesize vitamin C themselves, using a different biological pathway than plants. This feature has later been secondarily lost (e.g. humans and teleost fish) and regained multiple times in different taxa. However, this is speculation on my part, and I don't know this literature well enought to say if there are studies on the origin of vitamin C synthesis in animals, or if the pathways of vitamin C synthesis in plants and animals are fundamentally related. A review by Smirnoff et al (2001) indicates that there are some similarities between the enzymes involved in the final stages of vitamin C synthesis:

The gene and/or cDNAs encoding GalLDH, GulLO, and AraLO have been isolated and described from several organisms including cauliflower, sweet potato (GalLDH, 54, 121), rat (GulLO, 70), and Saccharomyces cerevisiae (53). In addition, complete nucleotide coding sequences of the A. thaliana and Nicotiana tabacum GalLDH genes (Accession No. AB042279.1, AB024527.1), and the Candida albicans AraLO gene (Accession No. AF031228) have been submitted directly to GenBank and associated databases. The predicted amino acid sequences of these related proteins share substantial amounts of identity. The mature cauliflower GalLDH amino acid sequence shares 28% overall identity with the rat GulLO and 26% identity with the Candida albicans AraLO (using the J Hein method with PAM250 residue weight table).

However, this doesn't necissarily mean that the synthesis of vitamin C is ancestral to both plants and animals. I haven't been able to find any papers on the origin of vitamin C synthesis, and if there is evidence that it is ancestral to all modern life.

In either case, I think it could be useful for you to look closely at vitamins in animals or other organisms for possible examples of the type of processes you are looking for, since they are (by definition) essential nutrients that some animals can and others cannot synthesize. They should therefore be good potential targets to find similar processes as your RNA-example. Vitamin B12 is an extreme to the other end, since it is essential for most animals but can only be synthesized by bacteria (so the main source is through bacterial symbiosis).

It is also possible that I'm mistaken, and that synthesis of all vitamins is ancestral to modern life, and that there has only been subsequent loss of the ability to synthesize certain vitamins/nutrients. Helliwell et al (2013) mentions this as a possibility in relation to the RNA-world, but also points out that the pathways for biosynthesis often differs between taxa, which would suggest that there has indeed been secondary acquisition:

The fundamental nature of vitamins is illustrated by the close structural relation of many vitamins to nucleotides (Figure 1), reflecting the likelihood that they were present in the ancient RNA world. Yet, their biosynthetic pathways appear to have arisen via the patchwork model of pathway evolution [31], resulting in the recruitment of unrelated proteins in any one pathway. Indeed, many vitamins are synthesized by different routes in different organisms [20]. For example, alternative pathways are found for thiamine biosynthesis in prokaryotes and among different eukaryotes (Figure 2) [20]. Despite this diversity in biosynthetic pathways, there appear to be common trends in the causes and mechanisms underlying pathway loss.