The phrase "stopped clock" or "broken clock" comes from the proverb "even a stopped clock gives the right time twice a day." Someone, even a crank, denialist, or a conspiracy theorist can be correct once in a while, often by chance alone. This may be because predictions just happen to come true but also it can be the one exception to the rule of crank magnetism. I.e., a crank may find some crankery that is so crankerous that they can't help but denounce it as crank quite vocally.
This analogy is one counterpoint to the fallacy fallacy, which is the mistake of assuming that any conclusion derived from fallacious reasoning must be false. To extend the analogy a bit further, a clock stopped at 3:48 will always conclude that the current time is 3:48. Thanks to its faulty premises, though, this conclusion happens to be correct only for a total of two minutes out of any given day, or roughly 0.14% of the time.
Grigori Rasputin's magical cure for hemophilia? He instructed the emperor and empress to discontinue all medical treatment being received by the tsarevich. This included aspirin, a drug which definitely does not help hemophilia (it can cause minor stomach bleeding, and in cases of hemophilia, there's no such thing as "minor bleeding"). It's unlikely that Rasputin knew that asprin was worsening the condition, yet his advice probably prolonged the tsarevich's life.
In Aristotle's treatises, often the most obtuse points by other philosophers are addressed as a starting point for an issue. Usually this is done to consider an issue far beyond the way it is usually addressed, though to modern audiences, the style can seem extraordinarily artificial, like an extreme counterpoint. On the other hand, sometimes it is used subtly to record a complete history of thought on a particular matter without bothering the reader for not keeping up. Many of the citations Aristotle makes are the only surviving fragments of certain authors.
Petr Beckmann was both a Randroid and a believer in crank ideas about physics and engineering. However, despite this, he held no tolerance for those who tried to square the circle, and wrote a whole book about the foolishness of the idea.
 Source of the analogy
The term refers to an analogue clock with the standard twelve-hour dial face. "Flip digital" clocks (where the numbers are affixed to small plastic cards that "flip" into position when driven by an electric clock mechanism) can also be considered to be "right" twice a day. Such a clock set up for 24-hour time would still be right once a day. Clocks with LED or LCD displays that are stopped normally go blank, falsely implying that, for some bizarre reason, time no longer exists (hence the lack of time to keep track of) - or they flash repeatedly, implying that time has gone wibbly-wobbly-timey-wimey again.
 Objections to the analogy
Without a frame of reference, there is no way of knowing when a stopped clock is right... or not. So to determine the time that a stopped clock is correct you'd need another clock, and if you have another clock why bother with the stopped one? Plus, if you cannot tell when the stopped clock is right then it is worse than a clock displaying the wrong time which, so long as the timing mechanisms work, can at least inform you of the passage of time. Plus, plus, the concept of the 'right time' is subject to quite a bit of uncertainty - not least because of relativity. A stopped clock with a second hand is right twice a day to an uncertainty of one second, but technically is it only right for one planck time, a measurement so small as to be pointless. But this is why analogies are just that, analogies.