What are the minimum requirements for becoming pope? If the conclave decided for whatever reason to choose a person who was not overwhelmingly qualified such as a cardinal or other prominent bishop, what requirements need that person have fulfilled to be eligible for election?

I assume they have to be Catholic obviously, but any other requirements?
The Answer

You asked what were the minimum requirements for becoming a pope. So I will not entertain all the minutia associate with papal elections.

First of all, a pope is simply a bishop who has universal authority over the Church1. Other bishops only have authority over their particular Churches, though they may exercise authority over the universal Church if they are joined by other means (i.e., an oecumenical council or in union with another bishop).

Further, bishops are bishops by virtue of the fact that they are recipients of the sacrament of priesthood. To eligible to receive this sacrament, one must be a baptized man2.

Therefore, because the papacy is simply an office of a bishop (or episcopate) with universal authority, the only requirements for holding the office of the pope, the papacy, are the requirements for holding an office of a bishop. The only requirements for holding the office of a bishop are those required to receive the sacrament of the priesthood. Therefore there are only two requirements:


One must be male
One must be baptized


All other "requirements" are merely practical requirements--that is to say, it is very doubtful the Church will elect a non-practicing Christian despite the fact he was baptized as a baby and a male. Despite the fact he is eligible, he is practically disqualified on account of his apostasy.

Side Note

There is room for speculation in the Church on whether popes can lose their office save the occasions of death and resignation. St. Robert Bellarmine is infamous for speculating such. While this is in the realm of speculation, the Church in general has moved strongly away from such thoughts. It is popularly held that popes are popes until they aren't popes--that is to say, a pope doesn't stop being pope because you don't like him.

Sources

1http://www.scborromeo.org/ccc/para/882.htm

2http://www.scborromeo.org/ccc/para/1577.htm