From what I understand saints and missy M.(as the first saint) are not "worshipped" but "venerated". The distinction being that worship supposedly means acknowledging the object as a divine, a god. The venerated saints are to be held on the pedestal as the role models - the people who got the closest to the ideal virtues of a true Christian.
One is not supposed to go overboard with the veneration, as has been declared after one of the latest ecumenical councils.
It's a lot more complicated than "this was a good person". In order to be canonized, one must first be beatified. In order for this occur (on top of having had to have lived a good life, been Christian, etc.) a miracle must have occurred in some conjunction with the person (while alive or posthumous). This is taken as evidence that the particular person enjoys some special connection to God that the laiety lacks.
Then, in order to become a Saint, another miracle must occur after their death, by their intercession.
The general idea is that the second miracle proves that the deceased is already in Heaven (the rest of humanity having to wait until Judgement Day). Sainthood is the Church saying "We're pretty sure this person was soooo good that they got a GO STRAIGHT TO HEAVEN card." And because of that, and because they have a special connection to God, they can do things like grant miracles or intercede on behalf of mortals to God.
The notion of patron saints pretty much requires an Old Testment kind of God, IMHO.
God: "I will bring mighty storms over all the lands, for the nations of the Earth displease me with their iniquity!" (I've always liked the word iniquity...)
St. Andrew: "Ok, but could you maybe spare Scotland (and Russia, Ukraine, Barbados, Sicily, Greece, Cyprus, Romania, etc.)?"
God: "Uuuuuughhhhh. Okay, fine."
St. Dominic: "And spare the astronomers of the world."
God: "I...what?"
St. Valentine: "And the beekeepers!"
God: "FUCK THE BEEKEEPERS!"
And thus, colony collapse syndrome was created.
Seriously though....it's thinly-disguised
tutelary deities. Initially, sainthood started off as a way to recognize martyrs, and some weird backward association with the manner of their martyrdom became infused into the notion of "patron" sainthood. St. Bartholomew was flayed alive, so he became the patron saint of leatherworkers. St. Lawrence was grilled on an iron grate, so he became the patron saint of cooks.
Later, sainthood was given to those who had merely been devout without martyrdom ("confessors"). The association continued. St. Lidwina fell while ice skating and became deformed and started losing parts of her body the rest of her life, while also becoming a mystic and healer. BAM -- patron saint of the chronically ill *and* ice-skating.
Couple that with conversion efforts towards polytheists that were used to having various deities to pray to for different things, and it's a no-brainer.
"oh, you can still pray before smithing a sword, just to St. Clement (or Dunstan or Eloy) instead of Vulcan or Hephaestus or Branwen or whoever...."