While there are some philosophers today, known as "animalists," who identify the human person as their whole body (brain and all). many more philosophers hold that personal identity is a function of brain continuity. On this view, if your brain were transplanted into a different body, you would then be re-embodied in that new body. The way you phrase your question is interesting as you refer to "the body by its name" and refer to "the brain's name." This is rare, as (to take my own case) few would think that "Charles" refers to my brain or my body. I believe most would think "Charles" refers to me as a subject, a substantial individual being who thinks, acts, has feelings, has a past, and so on. It is a further question about what is essential (in bodily terms) for me, Charles, to endure over time. Would I survive if half my brain was removed? There is a recent book on such questions, Are we bodies or souls? by Richard Swinburne. Swinburne argues that reflection on personal identity should lead us to believe that we are more than our brains (and more than our whole animal bodies). Whether you agree with him or not, I commend his methodology: we should begin with a certainty about our own substantial existence as persons who endure over time. After such a beginning, we can then go on to ask whether you and I are brains or our continuity rests on our brain's identity, and so on.