At the moment, I don't have access to the early 1960's paper of John and Nirenberg that (from what I understand) introduced the space BMO (bounded mean oscillation). Why were John and Nirenberg interested in the space BMO? Were their interests motivated by PDEs? Did they initially make the connection with Hardy spaces? Any thoughts/input would be appreciated.
I do not know your location, but the article in question, "On Functions of Bounded Mean Oscillation" (1961), has been scanned and posted by Wiley and is available by subscription from some universities.
The original paper makes no direct mention of Hardy spaces or PDEs. For motivation, they cite a paper by John and a paper by Moser in the same issue of the journal, and they reprove a basic functional analysis result of Weiss and Zygmund. Fritz John's paper was an analysis of the polar decomposition of the derivative of a diffeomorphism of an n-dimensional shape. (I mean a polar decomposition in the sense of linear algebra: Every square real matrix is the product of a symmetric matrix and a rotation matrix.) What John had in mind was estimates for deformation strain of solid materials. (Interestingly, he had an Air Force grant. Maybe John or the grant officers were thinking of aircraft parts.) Both John's paper and John-Nirenberg have the flavor of geometric analysis and nuts-and-bolts Lp estimates.
Moser's paper establishes the Harnack inequality for solutions to elliptic PDEs. I suppose that Moser was the one who connected the new definition of BMO to PDEs. At the time, John and Nirenberg were at NYU, and Moser was either there or soon to arrive. So these were people in close correspondence.