Google Books is so cool. They have John Nystrom's 1862 proposal for a hexadecimal numeral system. Nystrom called it the "tonal system", because "ton" is the word for 16.
It's a pretty well-thought out system, with new numerals and base-16 addition and multiplication tables. Nystrom's ambition is pretty remarkable, and clearly a reason for the system's failure to gain traction. He proposes an entirely new system of words to replace the English number words, although perhaps there's more the replacing English's number words than he discusses. However, the number words he proposes as a system are actually pretty good. The words for numbers are generally simple CV syllables, and the words for the powers of 16 (16, 256, 4096...) are closed with a nasal consonant. The numeral system in general is similar to East Asian numbers, and the simple syllable structure prevents complex consonant clusters. "Vy" is the word for eleven, and "vytonvy" ("eleven 16s and eleven") is 187.
In general, this framework seems very elegant, and seems to point towards a framework easily extended to represent other bases. By adding a nasal consonant to the end of a CV numeral, you could establish that number as a base, as long as you developed words for the powers of that number. For example, "vyn" would be the basis for a base-11 "vynal system". There are problems with this system; Nystrom breaks the CV structure with the word for "one" and allows an initial nasal consonant, and the pronunciation he intends is somewhat unclear to me. I also suspect that the designs for the digits Nystrom chooses might be confusing (tho of course, that perception might simply reflect my inexperience with them). Adding some numbers for 17 to 20 could extend the system to cover vigesimal, allowing it to represent nearly any human numeral system.
I'm somewhat sympathetic, at least with mathematical thinking, to an weak inversion of the Sapir-Whorf hypothesis: that rather than an absence of words limiting the possibilities of human thought, the presence of words for counterintuitive concepts can be a "cognitive technology" that makes thinking in terms of those concepts easier. There's probably a much broader field of research on this than I'm aware of, but some relevant papers are "Number as a cognitive technology: Evidence from Pirahã language and cognition" and "Notation as a tool of thought".
Hex Headquarters has a system of English-inspired hexadecimal number words; Nystom's "vytonvy" (hexadecimal BB) would be "levtek eleven". Base42 is another hex system.