It was me that Mr. Jones thought could do it.
Is that required between thought and could?
English Language & Usage Stack Exchange is a question and answer site for linguists, etymologists, and serious English language enthusiasts. It only takes a minute to sign up.Sign up to join this community
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.