The Hebrew word naar נער has a number of meanings - two verbs and a noun. Let's look and see if they are related.
One verb means "to shake" or "to shake out." It appears 11 times with that meaning in the Bible. Sometimes the meaning is more than the simple "shake" as in Shemot 14:27, where it says that God "hurled" וַיְנַעֵר the Egyptians into the sea.
Here's Klein's entry for that meaning:
shake, shake out, shake off, stir.
— Qal - נָעַר he shook, shook out, shook off, stirred.
— Niph. - נִנְעַר 1 he shook himself free. 2 was shaken out; PBH 3 he bestirred himself; PBH 4 was poured out, was emptied.
— Pi. - נִעֵר 1 he shook out; 2 he stirred up.
— Pu. - נֻעַר NH 1 was shaken; PBH 2 was stirred; PBH 3 was emptied.
— Hith. - הִתְנַעֵר 1 he shook himself; NH 2 he bestirred himself.
— Hiph. - הִנְעִיר he encouraged. [Aram. נְעַר (= he shook, stirred), Syr. נְעַר (= he poured out), whence נָעוֹרָא (= waterwheel). Arab. nā‘ūra (= waterwheel with buckets, noria), is a Syr. loan word. cp. Arab. na‘ara (= it spurted, gushed forth — said of the blood of a vein), na‘āra (= earthen jug, pot).
Klein doesn't include it (perhaps it wasn't common in his time), but the hitpael form התנער hitnaer today means "to shirk" or "to renounce responsibility."
The other verb means "to bray, roar, growl" and only appears once in the Bible:
יַחְדָּו כַּכְּפִרִים יִשְׁאָגוּ נָעֲרוּ כְּגוֹרֵי אֲרָיוֹת
"Like lions, they roar together, they growl [na'aru] like lion cubs. (Yirmiyahu 51:38).
While in this verse the metaphor is for the growl of a lion, in Rabbinic Hebrew the verb was designated for the bray of a donkey, and so it continues today.
And here's what Klein writes about this meaning:
Aram.-Syr. נְעַר (= roared, growled, brayed), Arab. na‘ara (= rattled), Akka. nēru (= to growl), nā’iru (= roaring)
But by far the most common appearance of naar in the Bible is as a noun, meaning "boy, lad, youth", with sometimes the more specific sense of "servant" or "soldier." There are 240 occurrences with this meaning, and another 63 for the female form נערה na'ara (girl, maiden, servant.) Related words in Hebrew are noar נוער - "youth" and neurim נעורים - "adolescence."
So are any of these meanings related to each other? Klein does not connect the two verbs, but presents two theories as to the origin of the noun.
The first says that the noun, meaning "youth," comes from the verb meaning "to shake", which he extends to the sense "to throw":
The other theory connects it to the braying and roaring usage, as an "allusion to the roughness of the voice at the beginning of puberty."
One word that is nearly certainly unrelated to any of these is the Yiddish nar meaning "fool" (the source of the familiar Yiddish word narishkeit - "foolishness." The Yiddish nar derives from the German narr of the same meaning. The etymology of narr (or the related narre or narro) isn't clear. Some say it comes from the Latin naris, meaning "nose" (ultimately the source of the English "nasal"), developing from "sneering (with the nose)" to "mocking, jeering" to "fool." In any case, this word has been in German for a long time - which means that it's much more likely that Yiddish borrowed it from German instead of German borrowing it from Yiddish.
Our youth might need education, but we don't need to make them the source of all foolishness...