F. Cajori gives several pointers in his *A history of mathematical notations*, Vol. 2, page 40. I think (he's a bit unclear...) he attributes the notation to Kronecker, referring to Dickson's *History*, Vol. 3, page 93. Dickson, in turn, in page 138 of that volume, tells us that Kronecker uses that notation in [Sitzungsberichte Akad. d. Wissensch. (Berlin, 1885), Vol. II, p. 768-80]

He apparently had introduced numbers $F(d)$, $G(d)$, $E(d)$, and when he needed one more, he used $H$ :P

(Reading on, we find the first appearence of a lowercase $h$ in Dickson referring to a paper of Weber (Göttingen Nachr., 1893, 138--147, 263--4), so---since Dickson uses notation from the papers he is quoting, we can blame Weber for the change of case)