The pronunciation "みやげ" does not correspond to on'yomi nor kun'yomi of
土産, so I thought it was a gikun (義訓), but the combination of kanji
産 does not seem to provide the meaning of "souvenir" either. Based on the pronunciation, I previously thought that it was taken from the verb stem of
見上げる and the meaning "to look up at" does sound like it's related to giving souvenir, but the slight difference in the second syllables of "みやげ" and "みあげ" has a lot to say against this theory.
So what was the origin of the word and kanji