Functionality seems to be distributed in well-defined regions of the cortex, so you can lose fairly large chunks of brain and remain "the same person" (as assessed by yourself and others). If you lose your occipital cortex, for instance, you generally seem to remain the same person but acquire a disability (cortical blindness).
The anterior part of the frontal lobes does appear to be involved in "personality", in the sense of controlling the range of emotions individuals characteristically exhibit: a pattern that becomes recognizable to others and which is one feature of "personality". Hence the unfortunate vogue for Walter Freeman's prefrontal lobotomies, applied to people with various behavioural problems in the 40s and 50s.

The most famous prefrontal lobotomy patient in the world is probably Phineas Gage, who underwent the procedure under uncontrolled circumstances. While he was working as a railway construction foreman, an explosion sent a tamping iron straight through his head: there's a diagram of the extent and nature of of his injury here. He recovered consciousness almost immediately, and lived for another 12 years, but his personality had changed completely. Although this is often quoted as an example of the prefrontal cortex's role in the generation of personality, I've always felt there was a significant confounding factor involved: the guy had had a metal rod blown through his face and out the top of his head, for crying out loud! That might just change your outlook on life, all on its own.

Grant Hutchison