This is the earliest reference I have located:
Minkowski-Farkas Lemma in Banach Spaces, L. Hurwicz (1952).
The same result was also referred to as the Minkowski-Farkas-Weyl theorem in the 1950's, for example in
The strong Minkowski-Farkas-Weyl theorem for vector spaces over ordered fields, A. Charnes and W.W. Cooper (1958).
The three authors are associated with this result in view of the following contributions:
- H. Minkowski, Geometrie der Zahlen (Leipzig, 1896).
- J. Farkas, "Theorie der einfachen Ungleichungen," J. F. Reine u. Ang. Mat., 124, 1-27 (1902).
- H. Weyl, "Elementare Theorie der konvexen Polyeder," Comm. Helvet., 7, 290-306 (1935).