1540s, "senior pupil at a school charged with keeping order, etc.," from Latin monitor "one who reminds, admonishes, or checks," also "an overseer, instructor, guide, teacher," agent noun from monere "to remind, bring to (one's) recollection, tell (of); admonish, advise, warn, instruct, teach," from PIE *moneie- "to make think of, remind" (source also of Sanskrit manayati "to honor, respect," Old Avestan manaiia- "making think"), suffixed (causative) form of root *men- (1) "to think" (source also of Latin memini "I remember, I am mindful of," mens "mind") The notion is "one who or that which warns of faults or informs of duties."
The type of lizard (1826) was so called because it is fabled to give warning to man of Nile crocodiles. Meaning "squat, slow-moving type of ironclad warship" (1862) is from the name of the first vessel of this design, chosen by the inventor, Swedish-born U.S. engineer John Ericsson (1803-1889), because it was meant to "admonish" the Confederate leaders in the U.S. Civil War.
I now submit for your approbation a name for the floating battery at Green Point. The impregnable and aggressive character of this structure will admonish the leaders of the Southern Rebellion that the batteries on the banks of their rivers will no longer present barriers to the entrance of the Union forces. The iron-clad intruder will thus prove a severe monitor to those leaders. ... "Downing Street" will hardly view with indifference this last "Yankee notion," this monitor. ... On these and many similar grounds I propose to name the new battery Monitor. [Ericsson to Asst. Sec. of Navy, Jan. 20, 1862]
Broadcasting sense of "a device to continuously check on the technical quality of a transmission" (1931) led to special sense of "a TV screen displaying the picture from a particular camera."