"Hardly anything."

In their pre-rigorous versions of derivatives, Fermat used e, Newton used o, and Leibniz used dx (as did e.g. Euler). Seems like Lagrange then used i (possibly standing for “increment”?) a bit before 1800, as did Cauchy in the early 1820s. Lacroix’s introductory calculus textbook used k in French (1797) but his English translators used h instead (1816; this was an important textbook introducing Leibniz’s notation to English readers). There are other examples of h by the 1820s by e.g. Ohm. But there was by no means any strong conventional standard for at least another several decades beyond that. It’s not clear to me that it stands for anything in particular, beyond being a convenient letter that wasn’t a, b, c, d, f, or g, any of which might have been confusing.

You are right in that it is the Hermitian transpose.

Essentially, it is the standard transpose of a matrix, except in addition to swapping row/column indices, you also take the complex conjugate of the entry:

$$ a^H_{ij} = \bar{a}_{ji}$$

Gauss, in his Disquisitiones, used ad hoc notation for the class number when he needed it. He did not use h. Dirichlet used h for the class number in 1838 when he proved the class number formula for binary quadratic forms. I somewhat doubt that he was thinking of "Hauptform" in this connection - back then, the group structure was not as omnipresent as it is today, and the result that $Q^h$ is the principal form was known (and written additively), but did not play any role. Kummer, 10 years later, used H for the class number of the field of p-th roots of unity, and h for the class number of a subfield generated by Gaussiam periods (and "proved" that $h \mid H$); in the introduction he quotes Dirichlet's work on forms at length.

I checked the file with my pdf-viewer and got a message that this font cannot be displayed and is replaced. Then $10\%$ looked like $10\bullet\bullet$. It might be that your viewer displays it as $10 H$.

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