The shortest and most practical answer to "what does it mean to have a patron saint?" is:
In Orthodox families, and particularly in Orthodox countries a child's middle name is usually that of a Saint (that is, a recognized capital-"S" Saint). This name functions as a sort of "church name" or Christian name.
At certain times in church history, pagans who were being received into the church would have names given after gods, demons or pagan feasts, and they would change their name to reflect the new life they had found. This is analogous to God's giving new names to Abraham, Jacob, Paul, Peter, and perhaps, Levi-Matthew. To Orthodox people their name-day (the day on which their name saint is commemorated) is celebrated like a second birthday (although maybe more spiritual in the types of gifts).
In another way, a person's patron saint would be that member of "so great a cloud of witnesses" who surround us on whom that person would call for prayer. This isn't to say that the Orthodox believe Saints have the power to change our circumstances, but they do have the power to pray (some would say Revelation chapter 4 depicts that very scene), and since they're perfected in love we have every reason to believe that they would desire to.
Someone who converts as an adult might choose a patron saint whose life was particularly inspiring. St. Alexis (Toth) of Wilkes-Barre, PA was a byzantine-rite Roman Catholic priest who, after exploring the background of his people and the history surrounding why there are byzantine-rite Catholics, decided he needed to convert to the Orthodox Church, and he took his people with him. He spent many years ministering to the faithful of this country, especially encouraging those who were converting to Orthodoxy from other Christian traditions.
Of course, we didn't know any of that story at the time that we named Alexis, but he seems an appropriate patron for our whole family (see a more detailed life of St. Alexis, as well as his icon here).
Please understand that I say all of this as a novice, and as a wretched sinner besides that.
If you have any more questions about Orthodoxy, you could check out Our Life in Christ, an internet radio show done by two Protestant converts to Orthodoxy; or, here are links to the About Orthodoxy sections of the three main Orthodox jurisdictions within North America:
Orthodox Church in America
Antiochian Orthodox Christian Archdiocese of North America
Greek Orthodox Archdiocese of America
Thanks for your question, Nick!
Here is an icon of my patron St. Michael the Archangel (which was, by the way, already my middle name, just spelled wrong)