A bad penny always turns up
A disreputable or prodigal person will always return. More generally, this proverb refers to the recurrence of any unwanted event.
This proverb has lived long in the language. It derives from the notion that some coins were 'bad', that is, they were debased or counterfeit.
The 'clipping' of coins was rife in the Middle Ages, long before standardisation of the coinage was reliably enforced. This example from the reign of Edward I shows the degree of 'badness' that pennies then endured.
The term 'bad penny' was established enough in English by the late 14th century for it to have been used in William Langland's famous prose poem Piers Plowman:
Men may lykne letterid men... to a badde peny.
The expression continued to be used and, by the 18th century, the proverb as we now know it began to find its way into print. An example is found in 1742 in Henry Fielding's translation of Aristophanes Plutus:
This [the phrase ‘A very bad stamp’] is literal from the Greek... It was a Metaphor taken from their Money. We have a Proverb in English not unlike it, a bad Penny.
What made whoever coined this proverb link bad pennies with the notion of something unwelcome returning isn't now clear. Perhaps it was the sense that, if you clip or pass on a bad penny, it won't be long before it comes back to you in your change. Our present day expression of that would be 'what goes around comes around'.
See also: the List of Proverbs.