It was me that Mr. Jones thought could do it.
Is that required between thought and could?
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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.