In 1945, Grace Murray Hopper was working on the Harvard University Mark II Aiken Relay Calculator (a primitive computer).
On the 9th of September, 1945, when the machine was experiencing problems, an investigation showed that there was a moth trapped between the points of Relay #70, in Panel F.
The operators removed the moth and affixed it to the log. (See the picture above.) The entry reads: "First actual case of bug being found."
The word went out that they had "debugged" the machine and the term "debugging a computer program" was born.
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HOWEVER . . .
Well, the entry ("First actual case of bug being found.") shows that the term was already in use before the moth was discovered. Grace Hopper also reported that the term "bug" was used to describe problems in radar electronics during WWII.
The term was use during Thomas Edison's life to mean an industrial defect. And in Hawkin's New Catechism of Electricity, an 1896 electrical handbook from Theo. Audel & Co.) included the entry:
The term "bug" is used to a limited extent to designate any fault or trouble in the connections or working of electric apparatus.
In discussing the origin of the term, the book notes that the term is said to have originated in quadruplex telegraphy and have been transferred to all electric apparatus.
Common folk etymology says that the phrase "bugs in a telephone cable" was used to account for noisy lines. There is no support for this derivation.
However, the term "bug" was used in the early days of telegraphy. There were the older "manual" keyers that required the operator to code the dots and dashes. And there were the newer, semi-automatic keyers that would send a string of dots automatically. These semi-automatic keyers were called "bugs". One of the most common brands of these keyers, the Vibroplex, used (and still does use) a graphic of a beetle.
These semi-automatic "bugs" were very useful, but required both skill and experience to use. If you were not experienced, using such a "bug" would mean garbled Morse Code.
Radio technicians also used the term "bug" to describe a roach-shaped device consisting of a coil of wire with the two ends of wire sticking out and bent back to nearly touch each other. This device was used to look for radio emissions. This term "bug" was probably a predecessor to the modern use of "bug" to mean a covert monitoring or listening device.
But, lets go way, way back to Shakespeare. In Henry VI, part III, Act V, Scene II, King Edward says "So, lie thout there. Die though; and die our fear; For Warwick was a bug that fear'd us all."
Samuel Johnson's first dictionary includes a definition of "bug" to mean a fightful object.