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The Latin word mentula isn't properly defined in the Lewis & Short dictionary, but it does show up on Latin-Dictionary.net and Wiktionary. Both those dictionaries define mentula as "penis". But what's striking about this word is its resemblance to mens, mentis, which means "mind".
Are the two words related? Is mentula a diminutive of mens, as Wiktionary suggests? Are the two words descended from the same PIE root?
If indeed the two words are related, do we know why?
Well, this may obviously be outdated, but G.M. Messing banged out a 3-page treatment of "The Etymology of Lat. Mentula" for the Oct. 1956 Classical Philology (Vol. 51, No. 4, pp. 247–249).
His review of the scholarship to that point was
Lat. mentula 'membrum virile' has never been satisfactorily etymologized. Of the various suggestions made in the past none has been endorsed wholeheartedly by the etymological dictionaries of Walde-Hofmann and Ernout-Meillet.
Those suggestions were
Against these, Ernout/Meillet finds Zeuss's derivation "unconvincing"; Kretschmer found it "unobjectionable" but insufficiently "expressive" to account for the term's popularity; and Messing concurs with them.
Messing calls the supposed relation with Sanskrit "not improbable" but hard to justify owing to "phonetic difficulties" and the lack of other close cognates in Latin.
Walde/Hofmann calls Kretschmer's derivation "almost acceptable" and Messing finds it "ingenious" before going on to note its "faulty reasoning". Closer examination of the passage in Aristophanes causes him to take the pennyroyal as the pubic hair of the Thebeian plain, not the schlongs of the Thebeian women's lovers. He offers that Kretschmer's other references to flowery cunts similarly derive from analogy to pubic hair rather than resemblance to women's anatomy. He ends by questioning mint's supposed status as an aphrodisiac as reliant on cherry-picked examples, given its usual absence from ancient discussion of such medicines.
Messing's own submission is based on the Grecian example provided by E. Benveniste's "Formes et Sens de Μνάομαι" in Sprachgeschichte und Wortbedeutung (1954, pp. 13–18). Μνάομαι had previously been thought to consist of two homophones, one meaning "remember" or "mention" and the other meaning "woo" or "court". Benveniste established that the "second word" had actually arisen as a special figurative use of the primary sense. During that discussion, he noted how legal Latin was forced to use mentionem facere ("to make mention" meaning "to propose marriage") because a different derivation to mentior ("to think up" or "lie") disallowed Romans from turning mentio into a demonstrative verb. Latin similarly had to use an entirely different root (poscere, "to ask") for its word for "suitor".
Messing notes that Romance languages suggest an unattested low Latin *mentare ("to mention" or "to propose"), which would have permitted *mentulus ("dealing with proposals"). This would have then leant itself by the late Republican period to dirty jokes involving "wooing power" which explains why mentula isn't attested in early Latin and is first attested in Catullus, who preferred amusing and racy slang to straightforward vocab. He therefore proposes that mentula derives from some connection with the idea of the penis acting as the "suitor", "wooer", "courter" &c. of one's beloved. This is connected through mentio with the root of mens, but isn't a diminutive. Messing concludes:
Naturally, I should not dare insist on all the details of the etymology just proposed, for they are pure hypothesis. Nevertheless, I think that at least the comparison of mentula, sc. 'the suitor, the wooer,' with mentionem facere and *mentare is both reasonable and true to the obscenely humorous nuance which the ancients sensed in this term.
Against this, the chapter on "Mentula and its Synonyms" in J.N. Adams's 1982 Latin Sexual Vocabulary repeats Martial's appeal to tradition that mentula was in use at the time of King Numa and describes it as "the basic obscenity for the male organ", "certainly... not felt by Cicero to be metaphorical" in any way. To Cicero, it was "that word" and not dignified to write directly, even in his discussion of the lack of a diminutive for menta. In Adams's opinion, it's "scarcely conceivable that the spearmint stalk was so suggestive to so many people that the metaphor could have caught on in the whole community". He notes Pokorny offers no etymology; refers to Ernout/Meillet's coverage of the 'prominent' and Sanskrit etymologies without noting how dismissive they are of them; and refers to Chantraine for coverage of a possible connection with mens (Vol. III, p. 693), without discussion of its likelihood.
Ah. Found the passage in Chantraine:
1 μήδεα: n. pl., the male sex organ... Other forms: μέζεα..., sing..., μέδεα... Μήδεα sometimes means «urine»...
Etym.: Obscure. It would be necessary to establish the connections between the three forms of the word. One would think that the old and vulgar form would be μέδεα, with the doublet μέζεα which supposes a gemination of δ and an aspirated pronunciation... It isn't likely that μήδεα derives from μήδομαι, although... Spitzer, BSL 40, 1939, 47 (who recalls after Friedländer, Lat. mentula, if that word derives from mens?). One would find more likely the hypothesis of Wackernagel, Spr. Unt. 224, n. 1, who after Nauck, sees in μήδεα a euphemistic substitution (which goes with its presence in the [Odyssey]). If one searches for the etymology of μέδεα, neither μαδάω, nor μεστός is suitable. Pokorny 706, recalls Ir. mess (from *med-tu?) «gland, glanded» with the original sense of «inflated», etc. But the origin of mess seems to be quite different...
2 μήδεα: «thoughts, worries», cf. μήδομαι.
That doesn't say much about what Adams was talking about at all, although it perhaps suggests an original derivation from Greek or its etymon, where—like British 'goolies'—an original neuter plural in reference to the testicles or whole kit eventually got used—like 'gooly'—in Latin's 1st-declension singular to refer particularly to the dick itself.
J.T. Katz's 1998 "Testimonia Ritus Italici" in Harvard Studies in Classical Philology (Vol. 98, pp. 183–217)—principally focused on the implications of testis meaning both "witness" and "testicle"—also mentions in passing (1) that the Russian words мудрый ("wise") and муде ("testicle") also "probably" both derive from PIE *men- and (2) that M.L. West argues there was an "almost universal ancient conception according to which the marrow originates in the brain and descends from there through the spine... to become semen...", an identity that shows up as late as Shakespeare's All's Well that Ends Well.
Meanwhile, in the middle of M.L. Wagner's 1907 Lautlehre der Südsardischen Mundarten (pp. 12–3), he makes cold water all over such theories by just drawing a straight line from mentula to mingere and meiere ("to piss"), making it—and presumably also Gr. μήδεα—the equivalent of English "pisser".
Ok. That turned into a massive (albeit interesting) textwall.
TL;DR:
When the linguists have had their say we can perhaps ask the poets for their opinion, famously Goethe, mirroring the Graeco-Latin elegiac metre:
Gib mir statt »Der Schwanz« ein ander Wort, o Priapus,
Denn ich Deutscher, ich bin übel als Dichter geplagt.
Griechisch nennt ich dich φαλλος das klänge doch prächtig den Ohren,
Und lateinisch ist auch mentula leidlich ein Wort.
Mentula käme von mens, der Schwanz ist etwas von hinten,
Und nach hinten war mir niemals ein froher Genuß.
