Pull out all the stops
Make every possible effort.
The popular belief is that this phrase derives from the manner of construction of pipe organs. These instruments have stops to control the air flow through the pipes and pulling them out increases the musical volume. This seems to be the type of casual easy answer that is the hallmark of folk etymology. In this case, the popular belief isn't a fallacy but is in fact correct.
Prior to the introduction of pipe organs the word 'stop' had, in a musical context, been used to mean 'note' or 'key'. That usage is recorded as early as the late 16th century, as in this example from George Gascoigne's satire The Steele Glas, 1576:
"But sweeter soundes, of concorde, peace, and loue, Are out of tune, and iarre in euery stoppe."
Of course, 'notes' and 'keys' can't be pulled out. The word 'stop' later came to be used for the knobs that control the flow of air in pipe organs, by pushing them in or, more to the point here, pulling them out.
The first person to have used the phrase in a figurative, that is, non-organ related, sense appears to have been Matthew Arnold, in Essays in Criticism, 1865:
"Knowing how unpopular a task one is undertaking when one tries to pull out a few more stops in that... somewhat narrow-toned organ, the modern Englishman."