Today’s Word of the Day over on Twitter is, er, quite something...
Two things about that word that need pointing out. First, it’s a fairly nonsensical, nonstandard term. (But then you’d probably already guessed that.)
Coined by the poet Robert Southey in 1834, sinequanonniness has seemingly never been used since outside of reprints and commentaries on his work; the OED has no other record of it (although they do have a record for an equally eye-catching adjective, sinequanonical, first used by the poet Thomas Moore in 1816).
Secondly, yes, it does indeed derive from the Latin expression sine qua non.
Sine qua non literally means “without which, not”—which on its own doesn’t feel like it means or implies very much, arguably. It’s almost as if, without any context, that string of words is all but meaningless. In fact, you could say that context is the sine qua non of understanding what that means. (Y’see what we did there?)
In essence, that’s how sine qua non works. Although classed as and functioning in a sentence as a noun, it carries much more of an adjectival force: it implies that what is being referred to is utterly indispensable, or utterly necessary. Without it, whatever it is you’re talking about simply would not be, would not work, or would not come to pass.
Although properly used as a phrase in English, slam all those words together and attach to it the multipurpose –ness suffix (used to form abstract nouns of state or quality), and there you have it. Sinequanonniness—a word implying absolute necessity or indispensableness.