Here’s what Wikipedia says about the etymology of Upanishad:
Max Muller as well as Paul Deussen translate the word Upanishad in these verses as “secret doctrine”, Robert Hume translates it as “mystic meaning”, while Patrick Olivelle translates it as “hidden connections”.
I suspect, however, that while this etymology is correct, the sense of it is not. Typically, words in this semantic field that begin with upa- convey, not secrecy, but closeness, in both an emotional as well as physical sense. An upaṭṭhāka is an attendent or carer or nurse; an upāsikā is a close and dedicated follower.
The Ambaṭṭha Sutta illustrates this sense through dramatic movement. When the brahmin student Ambaṭṭha enters the Buddha’s hut together with a number of other students (an unusually inimate setting), he sticks out by refusing to sit down with the Buddha, but walks around or stands. As the dialogue proceeds, his antagonism to the Buddha and his people becomes apparent. Only when a thunderbolt-wielding yakkha appears—what? sometimes extreme measures are required!—does he become terrified and seek shelter from the Buddha. And here, at last, he is said to “sit down close to” the Buddha:
Atha kho ambaṭṭho māṇavo bhīto saṃviggo lomahaṭṭhajāto bhagavantaṃyeva tāṇaṃ gavesī bhagavantaṃyeva leṇaṃ gavesī bhagavantaṃyeva saraṇaṃ gavesī upanisīditvā bhagavantaṃ etadavoca:
Ambaṭṭha was terrified, shocked, and awestruck. Looking to the Buddha for shelter, protection, and refuge, he sat close to the Buddha and said:
The otherwise closely parallel passage in MN 35 Saccaka omits upanisīdati here, proving that it applies specifically to Ambaṭṭha and his rude behavior. Saccaka, of course, was no brahmin.
Since this passage follows on immediately from a question about the ancient lineage of Brahmin teachers, it is not unlikely that this is intended as a direct usurping of their authority. Those brahmin teachers of old are not the real Upanishad, the text implies, they cannot protect you. But the Buddha can.