My hope of locating the Hodgsons in a few locations in the English county of Durham was dashed when I found an online posting for the Durham and Northumberland signatories to a 1642 document known as the Protestation. Originally envisioned as a national petition that would include an oath to uphold the true reformed Protestant religion, what results is a kind of census of males age 18 and older, as well as a few female heads of household; it includes an indication of those “recussants” who refused to sign or otherwise were identified as Roman Catholic. The Durham collection names about 120 Hodgsons arrayed in 40 or so parishes and towns – even allowing for some overlap and duplication, this becomes a much larger range of possibilities than I had anticipated. Add to this the proliferation of other Hodgsons in Cumbria and Yorkshire, and the likelihood of a single Hodgson Quaker nexus evaporates.
Furthermore, since the Robert Hodgson in question would have been only 15 or so at the time of the Protestation, he would not have been enumerated. If Robert is a common family name, it is found in Hartlepoole, Sedgfield, Bernard Castle, Lamsley, Lainchester, Lumey, Hebburn (“papists”), Auckland St. Helen, Hamsterley diocese of Durham (two), and Whitworth.
Similarly, if Robert did indeed have a brother Ralph, that name is found in St. Oswald parish (Elvett St. Thomas), Billingham, West Auckland, Bernard Castle, St. Hellin Auckland chappellrie (where he is churchwarden), and Hamsterley (three, as well as two Anthonys).
Thus, overlap is found at Bernard Castle, Auckland (combining St. Hellin and West), and Hamsterley, especially. Whether any of these possibilities bears fruit remains to be seen.
Orphan George 
Ah, it’s always a bit sad when an anticipated research path doesn’t pan out. Good to hear that at least you have more avenues to explore.