Idiom whose etymology involves misunderstanding the original meaning
Solution 1:
The exception that proves the rule is a good example. According to Wikipedia, based on Fowler’s Modern English Usage, the phrase has its origin in Roman legal doctrine, and at full length reads:
Exceptio probat regulam in casibus non exceptis or
The exception proves the rule in cases not excepted.
For instance, though not matter for a major legal case, suppose you see a sign at a stately home saying “sorry, we’re closed to the public today for a special event.” You would take this as an exception proving the rule that the stately home is generally open to the public.
Fowler’s example quoted in Wikipedia has more legal flavour:
Special leave is given for men to be out of barracks tonight till 11.00 p.m.; "The exception proves the rule" means that this special leave implies a rule requiring men, except when an exception is made, to be in earlier.
Current usage is more along the lines: “I always keep my New Year resolutions.” “Well, last year you said you would quit smoking, and you didn’t.” “Oh, that’s the exception that proves the rule.” This is obvious nonsense: this "exception" disproves the rule. Even if the claim had only been "I generally keep my New Year resolutions," an "exception" would at best not refute the rule, but never prove it.
So, in the original meaning, you realise something is an exception, and from the exception you infer there is a rule. The exception is therefore a case to which the rule does not apply. In current usage the exception is a violation of a rule previously stated. And then people nonsensically use the phrase to dismiss the contradictory evidence. In the worst cases people may even believe that a rule is proved true when found violated.
The exception that proves the rule is the topic of this question.
Solution 2:
I'm sure there are quite a few, but since we're familiar with their present meaning, it's kind of hard to know what they meant before... but here are two (I think).
Birthday suit - while I'm not sure when it began as an idiom (it was literally fancy clothes one wore on one's birthday (or the king's birthday, or some such) now refers to the clothes one was born with, i.e. the condition of being stark naked.
A close shave used to refer to miserliness (because a close shave by a barber meant you could go longer without another shave), whereas now it refers to a thin margin, a small margin of error where injury/damage could just as well have been done.
Solution 3:
A rolling stone gathers no moss
Rolling stones used to be uncool earlier but they are cool now.
...the original intent of the proverb saw the growth of moss as desirable, and that the intent was to condemn mobility as unprofitable.
https://en.wikipedia.org/wiki/A_rolling_stone_gathers_no_moss