Para Brahma Upanishad describes inner meaning of sacred thread (up-nayana / Janoi)
3. Then to this realized person reality (i.e. true wisdom) constitutes the inner tuft and sacred thread. To the Brahmana desirous of liberation is (allowed) the state of the inward tuft and sacred thread. The wearing of externally visible tuft and sacred thread (is necessary) for the householders engaged in rituals. The characteristic of the inward sacred thread is not clearly visible like external thread; it is the union with reality inwardly.
Interestingly, upnayana (up-nayana) means extra eye. Nayana means eye. It may be referred as third eye (of knowledge), yogic may call it 3rd eye as ajna chakra. 3 threads represents the famous trinity like satva-rajas-tamas, waking-dream-deep sleep, etc. Brahmin wearing 3 sacred threads means to rise above trinity of bondage. While the external thread (which is visible), makes a brahmin qualify to recite vedas, and upanishads, internal thread is more concerned with the union of reality i.e. knowledge of true Self - Atman / Brahman.