Assuming you're referring to the Han people, the people of Mainland China have called themselves "Han" Chinese since the Han Dynasty's reign. The Han was such an influential and important Dynasty in arts, literature, culture, government and society that people living in the Han Dynasty began to refer to themselves as such. Not only did it give the Han Chinese an identity, but it also made it easier to distinguish from other tribes that often pillaged ancient China (ie. Xiongnu, Qiang, Nanman). A majority (over 90%) of Chinese are Han, I haven't read anything that being Han over any other "type" of Chinese effected the Communist Party.

The CCP has affirmative action quotas for ethnic minorities, so on paper the ethnicity is not important to the participation of the CCP. The National Congress is also required to have delegates from each of the 56 peoples. Also remember that Deng Xiaoping, that veritable giant of the CCP, was ethnically Hakka.