The answer to this question comes from Dr Peter Evennett Hon FRMS.
In the early days, microscopists referred to lens aperture in terms of an angle expressed in degrees, sometimes corrected for the refractive index of the mounting medium (eg ‘balsam angle’)
As far as I am aware, and I'm not a ‘proper’ historian, the term ‘Numerical Aperture’, like so much else in microscopy, was introduced by Ernst Abbe. I've been looking through his fundamental paper ‘Beiträge zur Theorie des Mikroskops und der mikroskopishen Wahrnemung’, published in 1873 in M. Schultze’s Archiv für mikroskopische Anatomie, IX, 413-468. In this, when writing generally about aperture he uses the term ‘Oeffnungswinkel’ [ = opening (or aperture) angle ].
In his letter to the then Treasurer of the Royal Microscopical Society, J. W. Stephenson, which we have in our archive, written on 15 December 1876, Abbe uses the term ‘Numerical Aperture’. And in his description of his Apertometer, published in the Journal of the Royal Microscopical Society in 1878, and referring to a paper presented on 5 December 1877, he uses the term ‘numerical aperture’.
I'm thus almost certain that Abbe was the originator of the term, and that it dates from the mid-1870s
