So far I have been breaking the word into nara (नर) + Ayana (आयन) = home of nara (jiva). But lately I have realized I couldn't break it this way because to make it nArAyaNa (नारायण), nAra (नार) is needed instead of nara (नर). So what is the correct etymology of nArAyaNa? Was I correct anyhow?
I quote the following from the book Manusmriti:
He thought deeply, for he wished to emit various sorts of creatures from his own body; first he emitted the waters, and then he emitted his semen in them. That semen became the golden egg as bright as the sun with his thousand rays; Brahma himself, the grandfather of all people, was born in that egg. 'The waters are born of man' so it is said; indeed the waters are the children of the primordial man. And since they were his resting place in ancient time, there he is traditionally known as Narayana ('Resting on those born of man').
Narayana means resting on those born of man. Water is born of man. Therefore Narayana means resting on water.
The actual verse from Manusmriti:
Verse 1.10 [Meaning of the term ‘Nārāyaṇa’]
आपो नारा इति प्रोक्ता आपो वै नरसूनवः ।
ता यदस्यायनं पूर्वं तेन नारायणः स्मृतः ॥ १० ॥
āpo nārā iti proktā āpo vai narasūnavaḥ |
tā yadasyāyanaṃ pūrvaṃ tena nārāyaṇaḥ smṛtaḥ || 10 ||
Water is called ‘nara,’ — water being the offspring of nara; since water was the first thing created by (or, the original residence of) that being, he is, on that account, described as ‘nārāyaṇa.’ — (10)
(Sanskrit text, Unicode transliteration and English translation by Ganganath Jha)
Narayana consists of two words as Naara+Ayana. Naara means water and Ayana means home. It is mentioned in Mahabharata, Vana Parva, SECTION CLXXXVIII.
3 आपॊ नारा इति परॊक्ताः संज्ञा नाम कृतं मया
तेन नारायणॊ ऽसम्य उक्तॊ मम तद धययनं सदा
In ancient times I called the waters by the name of Nara; and because the waters have ever been my ayana or home, therefore have I been called Narayana (the water-homed).
There is website Digital Corpus of Sanskrit, that breaks the above shloka and give word by word meaning.
āpo nārā iti proktāḥ saṃjñānāma kṛtaṃ mayā / (3.1)
ap [n.p.m.] nāra [n.p.m.] iti [indecl.] pravac [PPP] saṃjñā [comp.]-nāman [n.s.n.] kṛ [PPP] mad [i.s.a.]
tena nārāyaṇo 'smyukto mama taddhyayanaṃ sadā // (3.2)
tad [i.s.n.] nārāyaṇa [n.s.m.] as [1. sg. Ind. Pr.]-vac [PPP] mad [g./o.s.a.] tad [n.s.n.]-hi [indecl.]-ayana [n.s.n.] sadā [indecl.]
You are correct -- in order to split nArayaNa, you cannot to nara + ayana. One of the correct split is na+ara+ayana. What doe this means? The word 'ara' means flaw (dOSha). When you have 'na' prefix it renders as 'absence of flaw'. Thus, the tatva which is 'ayana' (meaning aashraya/adobe) for flawlessness is called nArayaNa. Hence the philosophical doctrine of nArayaNa/vishNu being dOShavarjittaM (flawlessness) is core concept in Madhva's doctrine. There is a book by Sri.Jalahalli Sreenivasa Tirtha called 'nArayaNa shabdArtha' where some 76 different meanings for 'nArayaNa' word is expounded. Please read if you find a chance.
The Aryans didn't know the ocean where they started from - Central Asia. There are even some disputes whether "Samudra" means ocean. At any rate Narayana is absent from Rig Veda.
"ayana" doesn't mean resting place - the word for resting place is "ayatana".
"Narayana" is a Dravidian loan word into Sanskrit - the Dravidians were sea-faring from way back. Also in modern North India, "narayana" has changed to "narain" - showing that it is of a derivation that was strange to descendant of Vedic Aryans.
In the Tamil language, spoken in Tamil Nadu, the word tannir, which is tan (cool) plus nir (water) means cool water. In other Dravidian languages, spoken in south India, the word for water is nira, niru, or nir. But in Sanskrit, the language in which the early scriptures, the Vedas, were written, the words for water are apa or jala, which are completely different. Consequently, one might look to a Dravidian source for the origins of the story of Narayana on the waters.
Also in Tamil, the word ay means to lie down or to go to sleep, and the syllable an is a grammatical masculine ending; this gives the meaning for Narayana as “he lies down or sleeps on the waters.”
It would seem to make sense that the south of India, bordered on all sides by the ocean, might be the source of this evocative story of Narayana, who rises from the waters to re-create the world, after the mahapralaya, or great destruction.