I think this was first used in Eleanor Jorden's Beginning Japanese (1962). Here's how it's described on page 56 of her text:
Grammatically speaking, an extended predicate consists of a nominal n̄/no + √da predicate, with the nominal preceded by a sentence modifier.
Jorden was a prominent structuralist and student of Bernard Bloch, but I don't think Bloch ever used the term himself, and I don't think the term caught on with other structuralists or more generally in linguistics, although you might see it occasionally.
I think it's called an "extended predicate" because it adds another layer of predication to the clause:
〔〜だ〕＋のだ ＝ 〔〜な〕のだ
That doesn't really describe when it's used or what it means, but you'd be hard-pressed to come up with a label for のだ that does either of those things. A full definition simply won't fit in a short label. And that's okay—Labels Are Not Definitions.
Among both learners and linguists, in both English and Japanese, the most common term I've seen for this is simply のだ. That's also what you'll find it under in most dictionaries, and I think it's the closest you'll get to a standard name.
Sometimes this is written in katakana as ノダ in linguistics (to make the Use-Mention Distinction), as in the phrase ノダ文.