I didn't find anywhere why is the minute's kanji there in 自分? Is it because a meaning of 分 is "part". Please clarify.
分 isn't really the "minute's kanji", although that is one of its meanings. I believe the meaning of "part" came first though, and it is used for "minute" in the sense that a minute is a unit or part of time.
According to this source, the 分 from 自分 means the same thing as the 分 from from 本分, representing one's capacity/ability, and historically was used to mean the same as 私自身. The earliest recorded usage of the word 自分 was in 827AD.
If you do a search for 自分の分 you'll find discussion not of the etymology, but the philosophy of oneself, and how the definition of this varies from person to person. This makes me think it is still regarded with a sense of one's own part, abilities, responsibilities etc in life.