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The term Riesz basis is in wide use for the image of an orthonormal basis in a Hilbert space under a bounded invertible map (there are lots of equivalent definitions). But I've been unable to find any papers by either Riesz on this subject. Does anyone know a reference that justifies the name or otherwise know of a first use or something that sheds light on the origins of this term?

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According to the article Riesz system in the EoM, some early references that use this term are from N.K. Bari (or Bary). The 1946 reference seems hard to locate online, but the 1951 reference in Russian can be. I can confirm that she uses the terms Bessel system, Hilbert system, Fischer-Riesz system and Riesz basis without citing any other reference, except referring obliquely to her earlier work. It is conceivable that these terms were introduced by Bari. Also, in the first line of this 1951 comment by Gel'fand, he attributes the term Riesz basis to her.

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Barry, Welcome to MO. Let me translate for you the first sentence of Gelfand's paper: "N. K. Bari introduced a class of bases in a Hilbert space which she called Riesz bases." –  Alexandre Eremenko Jul 17 '13 at 19:47

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