The earliest Christian saints are martyrs, Christians killed for their beliefs. Because of Christians' ideas about resurrection, imitating Christ, and the afterlife, martyrs became like heroes in their communities. If the goal of Christian life is to be as much like Christ as possible, then surely dying in the manner of Christ - being executed for belief in Christ as the Messiah - was the ideal way to die. And if one lived and died as Christ, surely there was a reward to be had on the other side of death.

The common understanding in the Early Church was that Christians who died normally - not martyrs - would go to a place of waiting. Not an unpleasant place, but not the glory of heaven either. Just a calm place of refreshment, waiting for the Last Day and final divine judgement. Martyrs got to skip the waiting and go straight to the eternal reward of heaven. Their judgement had been passed at the moment of their death and there was no need to wait for the Last Day to be judged with the rest of God's creation. (There is also a belief that martyrs will somehow play a role in the judgement of others, but that's a little beyond the scope of the question, I think.)

Martyrdom was not something that Christians could seek out. No running to the local Roman governor or hostile Persians, shouting about Christ in hopes of getting executed. It was something to be prepared for, but never sought out. So when martyrs were executed, the people who had been their neighbours, friends, families, and fellow Christians understood it as both sad - having lost someone close to them - but also a victory, because that same person was now as close to God as a person can get. Of course, when we die, we leave bodies behind. Christians quickly adopted the practice of bringing the remains of martyrs home, treating them as precious artefacts that connected the community to the new resident of heaven. Chapter 18 of The Martyrdom of Polycarp describes this practice:

The centurion then, seeing the strife excited by the Jews, placed the body in the midst of the fire, and consumed it. Accordingly, we afterwards took up his bones, as being more precious than the most exquisite jewels, and more purified than gold, and deposited them in a fitting place, whither, being gathered together, as opportunity is allowed us, with joy and rejoicing, the Lord shall grant us to celebrate the anniversary of his martyrdom, both in memory of those who have already finished their course, and for the exercising and preparation of those yet to walk in their steps. (https://www.newadvent.org/fathers/0102.htm)

The places where saints' remains were kept became shrines and places of Christian worship. In this way, the earliest patron saints were patrons of particular places and of communities. Christians who had occasion to travel and encounter other communities who shared their faith were often keen to "meet" the local saints. The Pilgrimage of Egeria (spelled Etheria in some manuscripts) details this and several other interesting points about Christian worship of her era. (https://www.ccel.org/m/mcclure/etheria/etheria.htm) Churches are traditionally named for the saint whose relics (body parts, clothing, and other artefacts) are kept in the building. The practice of keeping relics in churches is not, today, as widespread as it once was and many churches are now named for the saint on whose festival the building is dedicated, whether there are relics present or not.

Because martyrs were understood to be residents of heaven but also connected to their earthly communities via their bodily remains, legends quickly rose about appearances by the martyrs and miracles that they worked. These were often related to healing of illnesses, sparing people from horrible fates, and protecting those who were in danger. Christlike examples from beyond the grave. This habit of miracle-working extended to saints who were not martyrs, a development that grew quickly when Christianity was legalized in the Roman Empire and opportunities for martyrdom were significantly fewer.

A saint's patronage is often connected to their earthly ministry, their manner of death, or the miracles they worked. For example, St Nicholas (the character associated with Christmas in many places) is reputed to have calmed a storm at sea to save the sailors, so he becomes a patron of sailors. He resurrects three boys who have met a ghastly Sweeney Todd style fate and provides dowries for three girls so that they can have good marriages. In this way, he becomes a patron of children, merchants, and brewers. (The boys' remains were hidden in beer casks and the girls' dowries were pouches of gold coins dropped down chimneys.)

While the comparison between Christian saints and their patronages and the spheres of influence of Greek and Roman deities seems similar, Christian saints have their origins in the lives and needs of local communities and, often, in identifiable real people. The legends about their miracles can become contentious and some saints are referred to as "legendary" because there is no historical data about them.

There are claims that a small number of Christian saints are pre-Christian deities who have been Christianized via syncretism. St Brigid of Kildare is probably the best-known saint in this category.

As a final note, saints and saints with patronages are not an exclusively Roman Catholic phenomenon. the Eastern Orthodox, Oriental Orthodox, Ethiopian, Anglican, and some Lutheran churches at the very least also maintain the veneration of saints with patronages. Their manner of veneration and commemoration and the dates associated with certain saints may vary, but it is a widespread Christian tradition.

Sources

Brown, Peter. The Cult of the Saints: Its Rise and Function in Latin Christianity

Higgins, Michael. Stalking the Holy: The Pursuit of Saint-Making

Moss, Candida. The Other Christs: Imitating Jesus in Ancient Christian Ideologies of Martyrdom

Anglican Church of Canada. For All the Saints