Right, there doesn't seem to be any authority (OEM, Merriam-Webster, American Heritage, etc.) that has any textual evidence of the true origin of this term. Not even Cecil Adams at *The Straight Dope" (here) dares claim that he can ferret a decent answer out.

My sense is that "fritz" is an electrification of "fizz" or "fizzle" in reference to the "fry" and "sizzle" of an electrical circuit partially or gradually shorting out.

Whatever the actual coinage, I believe that the widespread adoption of the phrase was almost certainly due to this onomatopoetic relationship (like "pffft!") to the fizzling out of various electrical commodities that were on the march toward universal distribution and familiarity right at the turn of the century. (OED cites a 1903 usage of "fritz" as the first in print, and Merriam-Webster 1902.)

The idiomatic phrase "on the outs" apparently does not appear in print until 1917 (ref. OED, though "at outs" appears much earlier), but a phrase like "on the wane" easily appears as early as 1857 and as it appears across all OED quotations starts to cluster around the turn of the century, indicating that the basic syntactical construction of "on the fritz" was well in place such that a transformation from "on the wane" to "outs" or "down" or "fritz" is at least plausible.

As some Internet articles point out, the actual OED quotation is in fact in reference to something to do with something "theatrical," specifically an "open air [performance] that put our opera house show on the Fritz." But as I read this quotation, I think it is silly not to see it as quite likely of metaphorical construction, primarily. Think about what happens when a stage play fails, or ends -- yes, indeed, the lights go out. By about 1890, electric arc lamps in the United States were well on their way to becoming common "fixtures" in the theater, and until tungsten lamps were introduced in 1907, they almost certainly would have been equally commonly and perhaps even constantly "on the fritz."

So, in my opinion, a) the timing of the coinage, b) the earliest source reference itself, and c) the manner in which common usage flowed (in my childhood, almost always applied to radios, televisions, and other electrical appliances) all indicate an onomatopoetic electrification of "fizzle" in combination with a common construction either derived from or similar in derivation to "on the wane" or "on the outs."