Etymology of "merry"
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From Middle English mery, merie, mirie, myrie, murie, murȝe, from Old English meriġe, miriġe, myriġe, myreġe, myrġe (“pleasing, agreeable; pleasant, sweet, delightful; melodious”)
Etymology of "happy"
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From Middle English happy (“fortunate, happy”), perhaps an alteration of Middle English happyn, happen (“fortunate, happy”), from Old Norse heppinn (“fortunate, happy”); assimilated to be equivalent to hap (“chance, luck, fortune”) + -y. Compare also Icelandic heppinn (“lucky”), Scots happin (“fortunate, blessed”). See further at hap.
"merry" is the older word in English going back to Old English, whereas "happy" is a borrowing from Norse that occurred sometime in Middle English. At some point in Modern English, "happy" replaced "merry" as the preferred word.