If your notice how the Levites were chosen as the priests, it actually goes back to Pharaoh and the plague that God sent to kill the firstborn of every family. Since those who had blood on the doorframe from the Paschal lamb were ‘redeemed’ from the death, every family owed 'the God of Moses' their firstborn. Not only so but as God also spared them of the first born of their cattle and also did not destroy their crops etc, the idea of all the firstborn or first-fruits as ‘belonging to God’ arose. Therefore as God brought them out of Egypt and into a land to be His people, Israel owed the first-born and first fruit of everything to God.
The Lord also said to Moses, “I have taken the Levites from among the Israelites in place of the first male offspring of every Israelite woman. The Levites are mine, for all the firstborn are mine. When I struck down all the firstborn in Egypt, I set apart for myself every firstborn in Israel, whether human or animal. They are to be mine. I am the Lord.” (NIV Numbers 3:11-13)
Possibly to make it more organized, and also to better foreshadow the future priesthood of Christ by the calling of God, under the symbol of high priest, who offered atoning sacrifices for sin. God ordained that in instead of selecting each firstborn the whole tribe of Levi would be set apart as dedicated to God. The dedication has nothing to do with personal holiness or fitness for the job, the dedication is by calling. The idea is that only God can create the Priesthood. Aron and Christ are the only two properly called priesthood heads.
You may wonder why I say ‘only two’ when we know Aron’s sons were included in the priesthood. Well that is because Aron made an imperfect shadow of the future reality in Christ. Since Aron was not good enough to be a good shadow of Christ, in that he had to die, many priests came after him, representing the same single office. This is the highest reason of the priesthood, to prefigure Christ.
The ‘calling’ of the priesthood can be seen here:
This is why he has to offer sacrifices for his own sins, as well as for the sins of the people. And no one takes this honor on himself, but he receives it when called by God, just as Aaron was. In the same way, Christ did not take on himself the glory of becoming a high priest. But God said to him, “You are my Son; today I have become your Father.” (NIV Hebrews 5:3-5)
The idea that the old priesthood was imperfect partly because ‘the priests died’ but Christ’s priesthood is superior, as he ‘lives forever’, representing the true Priesthood (or true High Priest for under him all believers are priests under him) can be seen here:
Jesus has become the guarantor of a better covenant. Now there have been many of those priests, since death prevented them from continuing in office; but because Jesus lives forever, he has a permanent priesthood. (NIV Hebrews 7:22-24)
Conclusion: So ‘Why were Aron and his son only ones eligible for being a Priest?’ Because the priesthood represents Christ’s priesthood and this can only be derived by the ‘calling of God’, not from the dedications of men. It is as a symbol of Christ ‘redeeming the firstborn’ who are every person who has his blood spread on the doorframes of their heart.