The correct way to incorporate the quotation is A:
According to one source, " He [Abraham Lincoln] was a natural raconteur."
The double quotes are used in order to transfer the exact words of someone, so they need to be used before and after the phrase " He was a natural raconteur". The square brackets are usually used to add information that was not originally there, in order to clarify something. In this example, the writer wants to clarify that the personal pronoun "He" refers to Abraham Lincoln, so s/he adds his name using square brackets.
