Why is St. Michael the Archangel the patron saint of sickness?

Answer 1

I don't know the answer, but perhaps I can lend some insight as to why St. Michael is the patron saint of sick people. Having had several severe, painful, and extended illnesses myself, at a young age, the experiences caused me to lose my faith in God. I wanted nothing to do with a God who would let me suffer like I did. Note, however, that it was not actually through my own free will that I turned from God, but I turned from God because of my perception of who He was. The illnesses were powerful enough to unfairly warp my perception of God, thus driving me from Him. It was not truly the result of my own actions that made me do this. I've thought that St. Michael protects people who are in my position, defending them against evil because their suffering is so far beyond the limits of human endurance that the person cannot be truly treated as if he willingly turned from God.

Now for something a little freaky. I am involved in a support group for people who had similar experiences to mine, illnesses so severe that as an aside, it crushed the spirituality of few of the people I know from the group. These several of us have all had paranormal experiences right after the worst of our illnesses. Before the illnesses got really bad, nothing happened. Afterwards, things would turn on by themselves (witnessed by others), and a few swear that dark, shadowy beings began to appear near them. Not close to them, but near to them, as if testing how close they could get before the person's faith forced them back. For each, the experiences only lasted a few weeks to a month or so, before stopping entirely. I have no clue what caused the these experiences, but I can't help but think that our faith had been so destroyed by illness that we each had been left "open", and that St. Michael is there to make sure nothing can get to us despite that.

Answer 2

Michael is the patron saint of many causes. See the list below. Sometimes these patronages were assigned hundreds of years ago and the reasoning behind it has been lost.
against danger at sea
against temptations
ambulance drivers
artists
bakers
bankers
banking
barrel makers
battle
boatmen
coopers
dying people
emergency medical technicians
EMTs
fencing
Greek Air Force
greengrocers
grocers
haberdashers
hatmakers
hatters
holy death
knights
mariners
milleners
paramedics
paratroopers
police officers
radiologists
radiotherapists
sailors
security guards
sick people
soldiers
Spanish police officers
storms at sea
swordsmiths
watermen
England
Germany
Coimbatore, India, diocese of
Iligan, Philippines, diocese of
Mobile, Alabama, archdiocese of
Pensacola-Tallahassee, Florida, diocese of
Rouyn-Noranda, Québec, diocese of
San Angelo, Texas, diocese of
Seattle, Washington, archdiocese of
Sherbrooke, Canada, archdiocese of
Springfield, Massachusetts, diocese of
Toronto, Ontario, archdiocese of
Albenga, Italy
Argao, Cebu, Philippines
Basey, Samar, Philippines
Brecht, Belgium
Brendola, Italy
Brussels, Belgium
Cabo Rojo, Puerto Rico
Caltanissett, Sicily, Italy
Carpineto Sinello, Italy
Castel Madama, Italy
Cerveteri, Italy
Cornwall, England
Cuneo, Italy
Dormagen, Germany
Dunakeszi, Hungary
Gaby, Italy
Gravina, Italy
Iklin, Malta
London, England
Marcianise, Italy
Naranjito, Puerto Rico
Papua, New Guinea
Puebla, Mexico
Salgareda, Italy
San Miguel, Iloilo, Philippines
San Miguel de Allende, Mexico
Sant'Angelo Romano, Italy
Saracinesco, Italy
Sibenik, Croatia
Siegburg Abbey
Toronto, Ontario, city of
Umbria, Italy
Vallinfreda, Italy
Zeitz, Germany
Congregation of Saint Michael the Archangel