Hugging is a universal form of endearment and the verb hug is a very common word in English, yet the origin of the word is unknown.
OED boldy says that "Appears late in 16th cent.: origin unknown." for its etymology, and adds that it shouldn't be mixed up with the word hugge:
Not to be confounded with hugge v. to dread, shudder, shrink with fear or cold. Not connected with Swedish huka, Danish sidde paa huk to squat. In some shades of meaning it approaches German hegen to foster, cherish, originally to enclose or encompass with a hedge; but it is difficult to see how they can be connected.
Although, hugge is an older form of the verb hug, used in 1500s–1600s.
Per OED, the earliest citation of hug (first used as hugge) is from 1567:
And hugge, and, busse, and cull, and cusse Thy darling apishe fruite.
1567 T. Drant tr. Horace Arte of Poetrie sig. Bvjv
With the form hug, though in past tense, it first appears in Shakespeare's Richard III as "He hugd me in his armes." in 1597, per OED.
The noun hug is from the verb hug; and the earliest citation is from 1659 per OED.
Etymonline also says of unknown origin for hug but adds some possible Germanic connections:
1560s, hugge "to embrace, clasp with the arms," of unknown origin; perhaps from Old Norse hugga "to comfort," from hugr "courage, mood," from Proto-Germanic *hugjan, related to Old English hycgan "to think, consider," Gothic hugs "mind, soul, thought," and the proper name Hugh. Others have noted the similarity in some senses to German hegen "to foster, cherish," originally "to enclose with a hedge." Related: Hugged; hugging.
It is not usual that OED completely rejects any connection and doesn't provide any possible routes, especially for common words (although there are other examples). Etymonline adds some possible connections but it is still not certain for a very common word like hug. Would it be possible to find further explanation or more details about the etymology of hug to unravel its mysterious origin?