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?
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:
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.
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.
What are the minimum requirements for becoming pope?
Technically any baptized male can be elected Pope.
The Code of Canon Law (1983) states:
Can. 332 §1. The Roman Pontiff obtains full and supreme power in the Church by his acceptance of legitimate election together with episcopal consecration. Therefore, a person elected to the supreme pontificate who is marked with episcopal character obtains this power from the moment of acceptance. If the person elected lacks episcopal character, however, he is to be ordained a bishop immediately.
One who is not yet a bishop (and the Church has elected several non-bishops to the papacy) can accept election, but must be immediately consecrated bishop. By implication, that would seem to require that a papabile (a) be male,(b) be baptized, (c) and be willing ordained deacon, priest, and bishop if necessary, and (d) have the use of reason in order to accept election and, if necessary, holy orders.
Let us keep in mind that the Prince of the Apostles, St. Peter himself was married and that Rome has named widowers as bishops in the past (Bishop Jean-François de Hercé of Nantes comes to mind).
The Holy Spirit inspires as He see fit!
*See Pope Paul IV's bull (encyclical) Cum ex apostolatus officio on how a heretical papal candidate would be invalidly elected.
source: this answer to the question In Catholic understanding when there is more than one Pope, who decides which Pope is real and by what criteria?
All those who are not impeded by divine law or by an invalidating ecclesiastical law are validly eligible [to be elected pope]. Wherefore, a male who enjoys use of reason sufficient to accept election and exercise jurisdiction, and who is a true member of the Church can be validly elected, even though he be only a layman. Excluded as incapable of valid election, however, are all women, children who have not yet arrived at the age of discretion, those afflicted with habitual insanity, heretics and schismatics.
