What should we call the clause after "would rather"?

Question

I’d rather you stayed at home tonight.

What should we call the clause after "would rather"? Are they "object clause"?

Answer 1

(lots of jargon coming your way)

I’d rather you stayed at home tonight.

The whole shebang is called a comparative construction, or a "term comparison" construction. The first (primary) term is "you stayed at home tonight", the second (secondary) term is implied, as in your sentence, or stated overtly:

I'd rather [you stayed at home tonight] than [that you went to the pub].

The part in the first block brackets, [you stayed at home tonight] is a "finite complement". It's a finite clause serving as a complement to the modal idiom "would rather".

It could also be called "a modal preterite clause" because the preterite (past) form of the verb ("stayed") is used not in the literal but in the modal sense, expressing wish: the action itself is positioned in the future ("tonight").

In the end, I'm not sure if it's an object clause. It could be an object clause:

I want [a cookie]. (Subject - Verb - Object)
I would rather [you stayed at home tonight]. (Subject - Modal - Object)

But I'm not sure. I'll wait for someone to answer. I'm not sure whether it's you that is the object or the whole clause in the brackets. My guess it that the whole clause in the brackets is the object, and you is the subject inside it, and then it must be an "object clause" indeed.

Answer 2

In my view it is an object clause with "that" omitted and in subjunctive mood.