Before understanding the meaning of “Tryambaka” त्र्यम्बक, let us consider the following Rica रिचा, श्लोक (incidentally one of the most powerful mantra-s) from the RgVeda:
RV 2.41.16 (to Devi Saraswati)
अम्बि॑ऽतमे । नदी॑ऽतमे । देवि॑ऽतमे । सर॑स्वति । अ॒प्र॒श॒स्ताःऽइ॑व । स्म॒सि॒ । प्रऽश॑स्तिम् । अ॒म्ब॒ । नः॒ । कृ॒धि॒ ॥
ambitame nadītame devitame sarasvati | apraśastā iva smasi praśastim amba nas kṛdhi ||
Translation: “Among all mothers (ambi-s), among all rivers, among all Devis, O Saraswati, thou are the best! O mother (amba), make us – the ignoble- full of renown.”
Here, both ambi अम्बि and amba अम्ब have been used to mean “mother”. So, amba means: mother -common noun- not Mother, but mother; and much more than that:
“womb” can be seen as coming from amba.
It also denotes the “waters” realm of Mother (Antariksha; Space).
That’s where ambā (“watery”/ “spacious”) comes from, as well as ambikā (ambi+kā; fem.).
Hence, ambuja (ambu+ja) means: “water-born”; ambar (amba+ra) means: antariksha, ākāsha.
And that’s why ambak (amba+ka; masc.) means: “of mother”. This is the only meaning Veda knows of, verily, as in this exquisite Rica (the famed Mahāmrityunjaya Mantra) for Rudra.
त्र्य॑म्बकम् । य॒जा॒म॒हे॒ । सु॒गन्धि॑म् । पु॒ष्टि॒ऽवर्ध॑नम् । उ॒र्वा॒रु॒कम्ऽइ॑व । बन्ध॑नात् । मृ॒त्योः । मु॒क्षी॒य॒ । मा । अ॒मृता॑त् ॥
tryambakam | yajāmahe | sugandhim | puṣṭi-vardhanam | urvārukam-iva | bandhanāt | mṛtyoḥ | mukṣīya | mā | amṛtāt
Translation: “I worship thee, the Three-mothered (Tryambaka), of pleasant odour, the giver of good health. Liberate me unto Immortality even as the gourd fruit is freed of its shell (i.e., when death comes).”
So, त्र्यम्बक = त्रि + अम्बक, Tryambaka = tri (three) + ambaka (of mother)
Tryambaka, then means, “of three mothers”. Rudra has three mothers: ILa, DakshiNā and Saraswati, when He is seen to come/ take birth evolutionarily from the Antariksha realm.
This realisation has been lost on the people since the end of the Vedic age, and upon much speculation and groping in the dark, Tryambaka was arbitrarily assigned the meaning: Three eyed. That even while Trilochana त्रिलोचन already means “three eyed” and is used very widely as an attribute of Shiva/ Rudra. However,
1) No etymology (in Sanskrit) exists for ambak meaning eye, in the same way that ambak is shown here to be from amba.
2) “Eye” (लोचन, चक्षु, नयन, etc.) is a utility word and therefore ambaka if meaning “eye” should have usage in regular contexts. This also doesn’t seem to be the case. However, in a classical text on Ayurveda, Ashtānga Hridayam, a word “valāmbaka”(vala+ambaka) वलाम्बकः occurs, and is said to be the place where the “remaining disease (phlegm)” is confined. Now, both Vala and ambaka are Rgvedic words. Vala is the Asat of Antariksha (Mother’s) realm who holds up Her waters, that is, is a barrier between us and Mother. The disease, in question, is being sent to this mysterious place, outside the body to “above it”, via the sahasradhara chakra, presumably. “valāmbaka” (Mother’s hold/ cave) is that place where this is sent and held up. This is how healing and self-healing is performed by Yogis. So here again, ambaka means “of mother” and not “eye” as translated by some.
3) There is this village named “ambaka” in Maharashtra/ India 17°12’4″N 74°22’25″E (wikimapia link). The site says the village has in its middle an ancient Ambā Temple, “from which it derives its name”. Moreover, the villagers must have been extremely conservative about this, given the fact that they fell under islamic dominance and surrounding areas do seem to have adopted islamic names. Now this is some hard evidence.