Etymology
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minute (adj.)

mid-15c., "chopped small," from Latin minutus "little, small, minute," past participle of minuere "to lessen, diminish" (from PIE root *mei- (2) "small"). Meaning "very small in size or degree, diminutive or limited, petty" is attested from late 15c. That of "particular, closely precise or exact" is from 1680s. Related: Minutely; minuteness.

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minute (n.)

"sixtieth part of an hour or degree of a circle," late 14c., from Old French minut (13c.) or directly from Medieval Latin minuta "minute of time; short note," from Latin minuta "a small portion or piece," noun use of fem. of minutus "little, small, minute," past participle of minuere "to lessen, diminish" (from PIE root *mei- (2) "small").

In Medieval Latin, pars minuta prima "first small part" was used by mathematician Ptolemy for one-sixtieth of a circle, later one-sixtieth of an hour (next in order was secunda minuta, which became second (n.)). German Minute, Dutch minuut also are from French. Used vaguely for "short time" from late 14c. As a measure expressing distance (travel time) by 1886. Minute hand "hand which indicates the minutes on a clock or watch" is attested from 1726. Minute-jumper (1890) was the name for the kind of electric clock on which the hands move only at the end of each minute.

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last (adj.)

c. 1200, "latest, final, following all others," a contraction of Old English latost (adj.) "slowest, latest," superlative of læt (see late); in some uses from late (adv.). Cognate with Old Frisian lest, Dutch laatst, Old High German laggost, German letzt.

Meaning "last in space, furthest, most remote" is from late 14c.; meaning "most unlikely or unsuitable" is from mid-15c. Meaning "most recent, next before the present" (as in last night, last September) is from late 14c.; latest would be more correct, but idiom rules and the last time I saw her might mean the most recent time this hour or the final time forever.

The biblical last days ("belonging to the end") is attested from late 14c. Last hurrah is from the title of Edwin O'Connor's 1956 novel. Last word "final, definitive statement" is from 1650s. A dying person's last words so called by 1740. As an adjective, last-minute attested from 1913. Last-chance (adj.) is from 1962. Expression if it's the last thing I do, expressing strong determination, is attested by 1905.

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last (n.1)

"wooden model of a human foot used by shoemakers," from Old English læste "shoemaker's last," earlier last "track, footprint, footstep, trace," from Proto-Germanic *laisti- (source also of Old Norse leistr "the foot," Middle Dutch, Dutch leest "form, model, last," Old High German leist "track, footprint," German Leisten "last," Gothic laistjan "to follow"), related to Old English læran "to teach," from PIE root *lois- "furrow, track."

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last (n.2)
late Old English, "the last or final man, object, time, etc.," from last (adj.). From late 14c. as "most recent person, latest comer." Also in Middle English as a noun, "duration" (early 14c.), from the verb. Phrase at (the) last is from c. 1200; extended form long last is from 1520s. To the last is from c. 1400.
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last (adv.)
c. 1200, "most recently;" early 13c., "finally, after all others" (contrasted to first), contraction of Old English lætest (adv.), superlative of late (see late).
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last (v.)

"endure, go on existing," from Old English læstan "to continue, endure," earlier "follow (a leader), accomplish, carry out, perform," from Proto-Germanic *laistjan "to follow a track" (source also of Gothic laistjan "to follow after," Old Frisian lasta "to fulfill, to pay (duties)," German leisten "to perform, achieve, afford"), from PIE root *lois- "furrow, track." It is related to last (n.1), but not to last (adj.). Related: Lasted; lasting.

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last-ditch (adj.)

"on the last line of defense," 1909, from an image attested by 1715, from a quote attributed to William of Orange (1650-1702), who is said to have uttered it defiantly during the French invasion of 1672; if so, originally in a Netherlands context.

We have no space to enter into the detail of the heroic struggle maintained by the young stadtholder and his faithful Dutchmen; how they laid their country under water, and successfully kept the powerful invader at bay. Once the contest seemed utterly hopeless. William was advised to compromise the matter, and yield up Holland as the conquest of Louis XIV. "No," replied he; "I mean to die in the last ditch." A speech alone sufficient to render his memory immortal. [Agnes Strickland, "Lives of the Queens of England," London, 1847]
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second (n.1)

"one-sixtieth of a minute of degree," also "sixtieth part of a minute of time," late 14c. in geometry and astronomy, seconde, from Old French seconde, from Medieval Latin secunda, short for secunda pars minuta "second diminished part," the result of the second division of the hour by sixty (the first being the "prime minute," now simply the minute), from Latin secunda, fem. of secundus "following, next in time or order" (see second (adj.)).

The second hand of a clock, the pointer indicating the passage of seconds, is attested by 1759.

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rpm 
1906, initialism (acronym) from revolutions per minute.
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