Is "which I thought was strange that" incorrect grammar?

Solution 1:

I agree that it's ungrammatical. People do say this kind of thing, but it's a mistake when they do.

You can say:

The family agreed, which I thought was strange.

This sentence has a non-restrictive relative clause modifying a finite proposition (The family agreed). Let's rearrange the sentence so that it has no relative clause:

I thought [that the family agreed was strange].

The clause that the family agreed was strange can usually be rearranged with an extraposed declarative finite clause (also possible is infinitival for the family to agree), and expletive it occupying the position normally reserved for the subject:

(I thought) it was strange that the family agreed.

Now consider the competing pair of clauses:

That a child so young could have so much influence was strange.

and...

It was strange that a child so young could have so much influence.

The writer has embedded in a relative clause a clause with strange, forgetting that the argument of strange is already in the main clause. A clause based on strange can have both extraposed subject and canonically-ordered versions, but the extraposed form is more frequent. In experience you expect to see an extraposed subject after strange, so the mistake is understandable.