A mysterious word skipped over in most etymological works of Old Icelandic is afr, áfr 'a beverage of some kind' (Zoega). It's not hard to see why not, there aren't any readily observed cognates in any other language - neither Indo-European nor otherwise. As far as I am aware, the only attempt to trace its origin is De Vries, who links it to a Proto-Germanic root *abara-; but research into De Vries' reasoning finds his etymology lacking, for reasons we will explore in a bit. But first, let's take a look at what clues we have in Icelandic for afr.
The word appears in the poetic Eddas of Icelandic lore, but the first attempt to provide a definition was by Árni Magnússon, writing a note on Egil's Saga that the word stands for sorbitio avanacaea, Latin for "oat drink." We see a glimpse of this in the Hárbarðsljóð poem, when Thor says
"...át ek í hvíld, áðr ek heiman fór, síldr ok hafra..." (verse 3)
"...I ate the rest before I went home, herring and porridge..."
Today's translations of Hárbarðsljóð write "porridge" based on the fact that a single copy of the poem has hafra (lit. "oats") instead of afra "oat drink." Cleasby dismisses this as a mistake: the oldest copies have afra. As afra went out of fashion, it is likely that hafra was inserted by a later copyist attempting to correct what he thought was a mistake. But the scribal confusion between hafra and afra is telling; reinforcing, though not proving, the idea that the drink was oat-based.
Finally, we cannot skip Modern Icelandic afir "buttermilk," which Cleasby notes was used in place of beer.
De Vries is the single scholar to attempt an etymology, focusing on the oats of the drink. Noting that Finnish apara "bierhefe" is probably not native, and likely a loan, he reconstructs Proto-Germanic *abara- with no definition posited. It would not impossible for *abara- to yield afr (with intermediate *apra- I presume) but *afra- is a conservative reconstruction (cf. OIc. hafr "goat" < PG *hafra-, Kroonen 2009).
A better etymology, I would argue, would be afr comes from Proto-Germanic *afra- and from there Proto-Indo-European *h2ep-ro-, that's *h2ep- "water" with the *-ro- noun-forming suffix that is prevalent in Proto-Germanic (cf. *wira- "man" < *u̯iH-ró-; *hafra- "goat" < *kap-ro-; *timbra- "timber" < *demH-ro-). As a root *h2ep- is well attested: Sanskrit áp- "wet;" Old Avestan āfš; Old Prussian ape- "river;" Tocharian B āp "water;" Greek Āpía "the Peloponnesus;" Oscan aapa "water;" Old Irish ab "river." Conspicuously missing, however, is a Germanic representative. While the phonological reconstruction is tight, evidence would suggest that the stem *h2ep- was lost or had changed to *h2ékw- (cf. PG *agra- "flood"). Some believe that *-kw- became *-p- (which is more likely than *-p- > *-kw-, though neither direction of change is rare), but I would argue that Anatolian evidence for unexpected variant *h2eb- points to an original bilabial stop, and that *h2ep-ro- was formed before palatization to *-kw-. The condition of cluster *-pr- preserved it from palatization.