Is
It was me that Mr. Jones thought could do it.
correct?
Is that required between thought and could?
Is
It was me that Mr. Jones thought could do it.
correct?
Is that required between thought and could?
I always thought only the references to inanimate objects use that or which, otherwise who should be used.
"It was me who Mr. Jones thought could do it." - sounds perfectly fine to me. ("Mr. Jones thought I could do it.")
P.S. I intended the text as a comment to an answer above, but cannot comment yet due to the reputation level.
With a few more changes I think this is even more natural:
I was the one who Mr. Jones thought could do it.
In my experience, a sentence like "It was me thought he could do it" sounds unusual, but (contrary to @johnlawler's statement) not particularly ungrammatical - especially if the word "me" is stressed.