Tolkien used similar forms throughout his life for Elvish words connected to “tears”, the most enduring being Q. Nienna “Lady of Pity and Mourning” and S. nirnaeth as in Nirnaeth Arnoediad “[Battle of] Tears Unnumbered”. The first manifestation of this root was as ᴱ√NYE(NE) “bleat” and ᴱ√NYEHE “weep” in the Qenya Lexicon of the 1910s, though Tolkien considered moving all the derivatives of ᴱ√NYE(NE) to ᴱ√NYEHE (QL/68). Nonetheless it seems the distinction survived in the contemporaneous Gnomish Lexicon which had both nı̯e and nı̯eχe as primitive forms (GL/59-60). Early derivatives include ᴱQ. nyé “bleat” and ᴱQ. nyéni “she-goat” along with ᴱQ. nyére/G. nîr “grief” and ᴱQ. nie/G. nîn “tear”, the last of these appearing in G. Nínin-Udathriol, the earliest name of S. Nirnaeth Arnoediad.
The root reappeared in The Etymologies of the 1930s as ᴹ√NEI̯ “tear” (Ety/NEI), though Tolkien considered and rejected alternate roots ᴹ√NEÑ (EtyAC/NEI) and ᴹ√NEI̯(ET) “moist” (NEI̯(ET)), the latter becoming the primitive word ᴹ✶neiti > ᴹQ. níte “moist, dewy” and N. nîd “damp, wet; tearful”. Other derivatives include ᴹQ. nie “tear”/N. nîn “tear” (same as the forms from the 1910s) as well as N. nírnaeth “lamentation” (Ety/NEI). Tolkien’s continued use of Q. Nienna (S/28) and S. nirnaeth (S/192) in later versions of the Silmarillion indicate this root’s ongoing validity.
Neo-Eldarin: For purposes of Neo-Eldarin, I think some of the Quenya nye- forms from the 1910s might be salvaged as abnormal vocalization from n(e)ye- < √NEY.
An unglossed root in The Etymologies of the 1930s appearing as both independent ᴹ√NEÑ (EtyAC/NEÑ-WI) and extended ᴹ√NEÑ-WI (Ety/NEÑ-WI), with derivatives like ᴹQ. nengwe/N. nem “nose” (Ety/NEÑ-WI). Similar forms meaning “nasal” appear in the first version of Tengwesta Qenderinwa from this same time period (TQ1: PE18/30, 46). This root is probably an updated version of primitive ᴹ✶nengǝ “beak, nose” in the Declension of Nouns from the early 1930s (PE21/26), and ᴱQ. nen (neng-) “nostril” from the 1920s is probably also related (PE14/72; PE15/75; PE16/113).