Impress Scottish friends around the world by pronouncing it as the Scots do: HUGMI’NAY (hug as in pug, ‘ preceds stressed syllable).
As some Scots do, anyway. The stress can be on the first or last syllable, or both; the first vowel can be as in “hog,” “hug,” or “hoe,” and the first syllable can even be “hang,” depending on where you live. The online DSL only has [hɔgmə′ne:] (hog-mə-NAY), but my hard-copy Concise Scots Dictionary has all the variants I mention. The DSL also has a good etymology:
[O.Sc. hagmonay, = 2., from 1604. The orig. of the word has been much disputed but the only satisfactory etym. is the derivation from North. Fr. dial. hoginane, with variants hoginono, hoguinettes, etc. from 16th c. Fr. aguillanneuf, a gift given at the New Year, a children’s cry for such a gift, New Year’s Eve, the second element of which appears to be l’an neuf, the New Year. Agui- is obscure (not < au gui). A similar development is found in Sp. aguinaldo, id. In Scot. the word is prob. due to the French Alliance and had been borrowed a.1560.]