The name Byggvir is usually traced back to the word bygg, which means barley. This leads to the idea that He may be a spirit or god of the crops and makes sense of Loki’s words, who says that Byggvir can be found under the quern-stone or hiding in the straw. As for Beyla, the etymology is less clear and several theories have been put forward: baula (cow), baun (bean) and biuilo (bee). The first of the three may mean that She’s a milk maid, thus explaining why Loki calls her “dirty”, but the same could be said of a bee or honey-making woman. In any case, the sense seems to be that of a rural setting or farmstead.
Personally, this reminds me of the folk tales about house elves or gnomes who keep things going and do everyday chores like milking animals and keeping the barn safe. That the written sources say virtually nothing about Them actually reinforces that idea: if Byggvir and Beyla are a sort of house elves, They would be absent from the major mythological narratives, but highly present in domestic cults. And if Lokasenna is a late poem by an Icelandic antiquarian – which is one possibility – then the author(s) could have added elements that were still present or remembered in a family environment in medieval Iceland.
This, however, is historical speculation. Among modern devotees, a common belief is that Byggvir and Beyla are two elves and my impression is that They resonate better with a rural household that has farming fields and cattle; still, there may be a more urban link in the use of flour, milk or honey in a domestic context. In any case, the notion is that of household deities under the authority of Ingui-Frey, which, again, points to a role of His as a god of the home.
Note:: illustrations by Thorskegga Thorn.