Etymology of "on the blink"

Solution 1:

The most persuasive discussion of the etymology of "on the blink" is this article that investigates the etymology of the American expression "on the fritz," for which it notes that the British and Australian equivalent is "on the blink."

A key part of the article quotes yet another article, which notes:

The phrase is now a common expression meaning that some mechanism is malfunctioning or broken. However, when it first appeared — about 1902 — it meant that something was in a bad way or bad condition. Early recorded examples refer to the poor state of some domestic affairs, the lack of success of a stage show, and an injured leg — not a machine or device in sight.

It goes on to observe:

Some people have suggested it might be an imitation of the pfzt noise that a faulty connection in an electrical machine might make, or the sound of a fuse blowing. This theory falls down because none of the early examples is connected with electrical devices, and the phrase pre-dates widespread use of electricity anyway.

The article finally reaches the conclusion:

I’ve gone around the houses, considered this theory and that, but come to no very definite conclusion. But the truth is that nobody really knows, nor now is ever likely to.

I would suggest that the expression might be connected with the fact that, when we blink, we stop seeing -- our vision effectively stops working. Something that's on the blink would then be in a state where it's not working, or not working fully.

Solution 2:

Christine Ammer, The American Heritage Dictionary of Idioms, second edition (2013) has this entry for "on the blink":

on the blink Also on the bum or fritz. Malfunctioning, out of order, broken, as in The TV was on the blink again, or You drive—our car's on the bum. The first of these slangy expressions dates from the late 1800s and possibly alludes to an electric light that flickers on and off ("blinks"); the second, from the same period, possibly is derived from bum in the sense of "a contemptible person"; the third, fritz, dating from about 1900, is of unknown origin.

In 1886, a demonstration project in Great Barrington, Massachusetts, had delivered electric lighting to a 4,000-foot stretch of the town's main street from a 500-volt AC source, so attributing "on the blink" to a flickering electric light is not necessarily anachronistic if the idiom came into use in the late 1800s, as Ammer says it did.

The oldest instance of "on the blink" in the relevant sense that I've been able to find is from 1896, antedating the OED's earliest citation (mentioned in the OP's question) by five years. From "Spoiled the 'Snap'," in the San Francisco [California] Call (August 23, 1896):

"One day George [a deputy clerk at "the office of the justice clerk" at San Francisco City Hall] thumped the button [on the office's telephone] and waited for Central to do the rest, but there was a hitch somewhere, and in spite of the fact that he had contributed nothing to the support of the corporation [that is, the central telephone exchange] he got impatient.

"He banged the hook for a while and when Central answered said:

"'Say, Central, this machine is on the blink.'

"'What's the matter?' asked Central. "'That's what I want to know,' said George. 'I guess she must be out of order.'

"He got his call and thought no more of the matter. The next day a repairer came from the telephone office, took out our soft snap and put in one that can't be fooled into deceiving Central. No more free messages."

Clearly the phrase "on the blink" is already being used idiomatically in this early instance.