This phrasing appears a lot. In a subject position, phrases like "a younger me" function as ordinary singular noun phrases rather than as pronouns.
First, from Wil McCarthy. "The Policeman's Daughter." Analog, vol. 125, iss. 6, June 2005, p. 8-34, as found on the Corpus of Contemporary American English.
Even a younger me, a green me fresh out of school, is better qualified than most attorneys to wrestle this particular alligator.
In this case, me (the first person pronoun in the objective case) is being treated as a noun referring to the speaker. The verb is corresponds to any singular third person noun phrase, rather than to a first person noun phrase, further signalling that this isn't a usual pronominal usage. Finally, it takes an indefinite article, specifying that this is one of several possible mes who fits the criterion of each adjective (younger, green). Such objectification works in a speculative fiction story that involves the technology of copying oneself.
Second, from Tom Ough. "Sir Richard Branson: 'If anything goes wrong, I put it out of my mind and move on.' " The Telegraph, 26 Nov. 2017.
A younger me might have been disappointed to know I’ve remained a dreadful driver.
Because the adjective modifies something about the speaker (younger, fitter, stronger), the usage commonly precedes modal verb phrases like "might have ..." and "would have ..." (News on the Web features several further results like this from credible publications.) Talking about oneself in this way allows a speaker to speculate about their performance without using a more verbose turn of phrase:
When I was younger, I might have been disappointed to know ...
A younger version of me might have been disappointed to know ...
I can't find the usage "a younger me" earlier than the 20th century (Google Books search). While adjective-me constructions are much older than that ("And make a conquest of unhappy me" is from Shakespeare), the me-as-reference-to-myself crops up in the 19th century (Thomas Carlyle wrote in 1828, "Haunted and blinded by some shadow of his own little Me"). Here's another example from Rudyard Kipling, "Divided Destinies," orig. 1886 (here from 1900):
An inscrutable Decree Makes thee a gleesome fleasome Thou, and me a wretched Me.
Note the switch to the nominative from "thee" to "thou" but the use of "me" both times. This specific usage is peculiar to "me."
Why "me" and not "I?" Dictionaries mainly indicate that I is unlikely to be encountered in this way. The Oxford English Dictionary, "me, pron.1, n., and adj." documents several uses of "me" with adjectives and various articles under "B. n." (the Shakespeare, Carlyle, and Kipling quotes are all from here). However, there is no specific entry for uses like "a younger me," perhaps because it is recent enough to not be documented by lexicographers.
Meanwhile, there appears to be even less on noun uses of "I" with adjectives - a few entries, a few examples, and none in the spirit of "a younger I." In the example sentences, the "a/an" article never appears, and the adjectives are primarily "other," showing a more restrictive pattern than examples of "me." So the OED suggests - in an absence of precedence - that "a younger I" wouldn't work.
Corpus searches provide further evidence of absence. "A younger I" doesn't turn up in COCA or NOW as a distinct noun phrase.
Why are "a younger I" and similar uses rare or nonexistent? It could be arbitrary. It could also be something about the case structure of the first person pronoun that noun-me comes from: speakers may use the objective case to create distance between I-as-speaker and me-as-addressee. Since in these examples "I" is the position of the speaker or writer, me creates a discontinuity between the speaker and the younger/green/better qualified/unhappy/wretched self. In other words, me provides further disambiguation for readers who would expect the speaker's own thoughts and actions to come in first-person "I" in the nominative case.