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I suspect the pronunciation of experiment with the "near" vowel arose either due to influence of the spelling (the pronunciation of "e" in contexts like this tends to be rather unpredictable), the influence of the pronunciation of the related word experience (where the "near" vowel is regular because of the occurence of unstressed "i" before another vowel in the next syllable), or some combination of both.

I suspect the pronunciation of experiment with the "near" vowel arose either due to influence of the spelling (the pronunciation of "e" in contexts like this tends to be rather unpredictable), the influence of the pronunciation of the related word experience (where the "near" vowel is regular because of the occurrence of unstressed "i" before another vowel in the next syllable), or some combination of both.

Here are my thoughts, guesses, and the (not very substantial) evidence that I have gathered.

The phonetic similarity of the vowels might also have contributed to the development and maintenance of the variation. (Some other words can be pronounced with either of these vowels, such as feral, query, inherent, coherent, adherent, hysteria—although in these words the "near" vowel is actually preferred by prescriptivists because the vowel occurs in a stressed penultimate syllable, or before unstressed "i" followed by another vowel—and (atmo)spheric(al), for which most prescriptive sources seem to prefer the short vowel, but the long vowel of "near" seems to be common, probably in large part because of influence from the related noun (atmo)sphere.)

Here are my thoughts, guesses, and the small amount of evidence that I have gathered.

The phonetic similarity of the vowels might also have contributed to the development and maintenance of the variation. (There are a number of other words of Greek or Latin origin spelled with "erV" (where "V" is any vowel letter) that show variation between these two vowels, such as feral, sclera, query, inherent, coherent, adherent, hysteria—although in these words the "near" vowel is actually preferred by prescriptivists because the vowel occurs in a stressed penultimate syllable, or before unstressed "i" followed by another vowel—and (atmo)spheric(al), for which most prescriptive sources seem to prefer the short vowel, but the long vowel of "near" seems to be common, probably in large part because of influence from the related noun (atmo)sphere.)

There seems to be some evidence of this pronunciation from the 19th century in the form of the spelling "expeeriment" that seems to have been used intentionally by some authors in dialogue to indicate the pronunciation with the "near" vowel.

There seems to be some evidence of this pronunciation from the 19th century in the form of the spelling "expeeriment" that seems to have been used intentionally by some authors in dialogue to indicate the pronunciation with the "near" vowel. The context suggests that the authors viewed it as a pronunciation that might be heard from "substandard" speakers.