The OED defines this useful phrase as meaning “a situation or statement has at last been understood; a person has reacted belatedly” and notes, “Originally used with allusion to the mechanism of a penny-in-the-slot machine.” The first citation is from The Daily Mirror in 1939; all the later cites are British as well.
As I say, it’s useful phrase. The closest American equivalent would be something like, “The lightbulb went on,” which, besides being clunky, lacks the apt imagined “click” of the penny equivalent.
In addition to the OED cites, this Google Ngram Viewer chart suggests the phrase is indeed of British origin. The red line indicates British usage, the blue American.
(There’s a fair amount of noise in the chart, emanating from references to actual pennies actually dropping.)
The chart indicates a steady rise in U.S. uses through 2008, and it appears to be continuing. In the New York Times, through 2006, the phrase almost always appeared as part of quote by a British or Canadian person. But there have been about fifteen uses of it by Times writers since then. Quite a few of them came from the pen of one person, Deb Amlen, who writes the “Wordplay” crossword puzzle blog. Clearly, pennies have to drop or the puzzle doesn’t get solved.
notoneoffbritishisms.com 
Perhaps it would help to have a Brit of a certain age comment here. I had heard this came from the days of the “Button A/Button B” pay telephones. My understanding is that the penny didn’t drop until you pressed Button A.
When the call was connected, you, the caller, could hear the person you called, but they could not hear you. Many of them adopted the habit of saying (as I might), “Orr here!” That way, you’d know you had reached the right party.
You then pressed Button A to drop the coin, turn on your microphone and converse. My understanding also is that the operator could hear the coins “dropping,” and could determine from the tone, which coins you’d inserted (I’ve heard this was true in the States as well).
If you got the wrong party, you would instead press Button B, and your coin would be returned so you could try again.
I’m slightly too young to have made phone calls when buttons A and B were around, but this sounds about right. Then again, I wondered if the phrase came from coin-operated lavatory cubicles, which also gave rise to the phrase “spend a penny”.
They still had these in NZ in the 1980’s. It took me a while to get the hang of it! I used to think people were just being rude until one of them told me to press button A – then the 10 cents dropped and I could talk to them!
The penny dropped in all sorts of machines: most notably public lavatories, when that big old pre-decimal penny was used to operate the doors.
Man, I really need to change up my colloquialisms.
Button A and Button B! I had forgotten!
The call was, of course, 4d.
Well done Mr Orr.
I don’t think this is equivalent to a ‘lightbulb moment’. For me, a lightbulb moment is when you have a brilliant idea, or the solution to a problem occurs to you. ‘The penny dropped’ implies that something you were too thick to work out before suddenly becomes clear. More like ‘Well, duh’.
I thought this phrase refers to the Victorian penny in the slot arcade machines that used to be in places like The Palace Pier in Brighton.
Given the first recorded date of 1939 the telephone kiosk seems a more likely source.
The BT Kiosk history pages has
The Ericsson coinbox (circa 1911) required the caller to insert coins one at a time under the direction of the operator, before a call was connected. With the introduction of the Button ‘A’ and ‘B’ coinboxes in 1925, callers could dial their own local calls, having inserted the correct fee.
==
So even before 1925 the penny had to drop.
I had always heard or used the expression ‘the penny dropped’ with adverbs “eventually” or “finally” to stress a passage of time, followed by a moment of realization.
The ‘penny’ is not free falling through air but clunking its way down an imagined mechanical device before arriving at its expected destination, the out tray. The mechanical nature of such a device meant that the anticipated ‘penny’ did not always ‘drop’ but could get stuck in the mechanism: similar to a thought process.
My understanding was that the ‘penny’ didn’t silently ‘drop’ into the collection hopper of a slot machine but was change due, announcing its presence with a defining metallic ding as it hit the coin out tray. The ‘penny’ in the pay phone would be a returned coin(s) and I vaguely recall pennies given as change in cigarette machines.
The imagery of “the penny dropped” is in similar vein to “hearing the cogs grind” when referring to someone thinking: a mechanical interpretation of a mental process.
The “penny finally dropped” is less a ‘light bulb’ or ‘eureka’ moment, more a ‘Road to Damascus’ type realization.
A feature of note to North American readers is the size of the coin in question. The pre-decimal (1971) UK ‘penny’ was a huge coin for a small denomination: bigger than a Canadian “toonie” and heavier due to its high copper content. Indeed, old pennies were often referred to as “coppers”. The plural “pennies” was used for a numerical quantity but for a cash value the plural “pence” was used. I digress.
Generations have come and gone since 1971 Decimalization so most Brits now would not be familiar with old pennies, or mechanical coin operated devices, but we like to hold on to our idioms.
Even fewer Americans share the above familiarities and their one-cent coin doesn’t carry the weight (pun intended) so I doubt if the phrase will gain much traction in the US.
Canada recently devalued its one-cent coin out of existence so I don’t envisage many ‘pennies dropping’ there either.
I’m not sure if it is strictly a regionalism, but in Eastern Massachusetts one sometimes hears the phrase: “light dawned on Marblehead,” which has the same meaning as “the penny dropped.”
Marblehead is shore-side town north of Boston.
I have always understood this to refer to old telephone systems with A and B buttons. On pre-decimal coinage in the UK, twopence and threepence were pronounced tuppence and throopence (or thruppence). My guess is that young UK citizens today have lost those pronunciations. To indicate that you were talking about new decimal pennies rather than old pennies after the changeover, the convention was (and still is) that you say two p and three p rather than tuppence and threepence. In Joseph Conrad’s “Heart of Darkness” there is a reference to “a two-penny-halfpenny river-steamboat “. That would have been pronounced tuppeny-haypenny, but I bet young Brits today might not know this. Compare Agatha Christie’s Tommy and Tuppence.
I first came across the phrase in a British book from the Thirties or Forties – probably either one of Leslie Charteris’s “Saint” books, or one of the “Tommy Hambledon” spy novels by “Manning Coles”.
The full phrase used there was, as closely as i recall, “Eventually the penny dropped and the mechanism emitted a faint creaking sound,” which suggests an arcade device.
Ah. I just encountered a page {https://www.phrases.org.uk/meanings/280900.html} that says the OED specifically references the arcade machines (with a tiny and not-very-easy-to-make-out-detail picture of one such).
I think the current expression in American is “It finally clicked.” Probably similar in origin, since that is the sound a coin makes in a phone before the connection is made.
In Brazil we use “a ficha caiu”, which translates to the same thing and comes from the the days of payphones.