Ultimate derivation unknown. Theories include the following.
Possibly derived originally from a reduplication of 実(Old Japanese mu, modern Japanese mi, “fruit”), from the way that peaches often grow in clusters. However, the vowel shift seems unlikely given regular Japanese phonetic shifts. In addition, most reduplicated terms in Japanese have the 頭高型(atamadaka-gata) pitch accent pattern, starting high and falling, which differs from the 平板型(heiban-gata) pitch accent pattern of this term.
Possibly cognate with Old Japanese百(momo, “hundred; lots”). However, this also has the 頭高型(atamadaka-gata) pitch accent pattern.
Possibly a reduplication of 毛(mo, “hair”), from the way that peaches are hairy. The term is spelled as 毛毛 in some ancient documents. However, 毛 was commonly used as man'yōgana for its phonetic value, in which cases its original Chinese meaning of hair is usually ignored.
None of the above possibilities seems very compelling. Given the archaeological evidence, this term probably originated before the Japanese ancestor population migrated to the Japanese archipelago.
Possibly related to 梅(ancient mume, modern ume, “Japanese apricot, Japanese plum”).