The best way to write the underlined part of the sentence is A. him and I.
So the correct sentence would read: "Since Mom knows that I don't like to share with him and I, she gave separate Nintendo DSi's to him and me for Christmas."
However, in this context, the phrase "with him and I" is still not entirely correct because "I" should be used as the subject of a verb. The corrected phrase would actually be "with him and me" when considering the grammatical structure of the sentence. So the most technically correct option based on your choices would be B. Leave as is, since "with my brother" (the antecedent) aligns with "me."
To clarify, the best overall phrase would be "with him and me," but since that's not an option listed, "Leave as is" is the best choice in this context.