Does chanting the phrase "bloody mary" 3 times while in a dark bathroom while looking at a mirror really do anything?

I just did it. No.

The Urban Legend and it's variations regarding Blood Mary:

The avenging spirit goes by many names: Bloody Mary, Bloody Bones, Hell Mary, Mary Worth, Mary Worthington, Mary Whales, Mary Johnson, Mary Lou, Mary Jane, Sally, Kathy, Agnes, Black Agnes, Aggie, Svarte Madame.

Summoning Mary requires the right chant. "I believe in Mary Worth" is the key phrase according to one version, but others require the shouting of "Kathy, come out!" or the repetition of "Bloody Mary" into the mirror as many times as the ritual demands. (Sometimes Bloody Mary gets more of a script and is summoned by calls of "Bloody Mary! I killed your baby!")

The precise requirements of the ritual vary. Some specify that the mirror must be illuminated by a single candle; in others, there must be a candle on each side. In some versions, the message to Mary is repeated by just one girl who is either a volunteer or one selected by the others to summon up the mirror-witch. The number of chants needed to fetch Mary also varies.

What the mirror-witch does upon arrival varies too. She may strike her summoner dead, drive her mad, or fiercely scratch her face. She may merely peer malevolently out through the mirror, or she may drag one of the girls back through it to live with her.

Origins:
The research into Bloody Mary goes back to 1978, when folklorist Janet Langlois published her essay on the legend. Belief in summoning the mirror-witch was even at that time widespread throughout the U.S.

Mary is summoned whenever squealing girls get together for a sleepover, but boys have been known to call on her too. She's said to be a witch who was executed a hundred years ago for plying the black arts, or a woman of more modern times who died in a local car accident in which her face was hideously mutilated.

Some confuse the mirror witch with Mary I of England, whom history remembers as "Bloody Mary." An expanded version of that confusion has it that this murdering British queen killed young girls so she could bathe in their blood to preserve her youthful appearance.

Mary I of England (1553-1558) was anything but a famed beauty terrified of losing her looks -- she was a matronly, fortyish woman who had about as much sense of style as a dust mop. The idea of her bathing in the blood of slaughtered virgins to preserve her loveliness is ludicrous. She came by the moniker "Bloody Mary" because she had a number of Protestants put to death during her reign, as she tried to re-establish Catholicism as the religion of the land after the reigns of her father (Henry VIII, he who married six wives over the course of his lifetime and established himself as the head of a new religion rather than tolerate the Pope's saying he couldn't divorce wife #1 to marry wife #2) and her brother (Edward VI, who ruled after Henry died but passed away himself at the age of 16). Mary was a devoutly religious woman who saw what she was doing as the saving of her subjects' souls from eternal damnation, and in those times -- as crazy as this sounds now -- the eternal wellbeing of a soul was deemed far more important than the comparatively fleeting life of a person. That bringing the country back to Catholicism would also safeguard her throne was also a major consideration.

Mary I was the half sister of Elizabeth I (1558-1603). Both were daughters of Henry VIII, but Mary's mother was Katherine of Aragon and Elizabeth's mother was Anne Boleyn. Elizabeth I became Queen when Mary died, and she reigned for many years, coming to be called "The Virgin Queen" because she never married.

Some muddlings of this "murdering queen" variant claim that Mary, Queen of Scots (1542-1567) is the "bloody Mary" of mirror summonings. Though this Mary was indeed a vain and foolish woman, history does not know her as a murderous one. (Well, okay, she did have a hand in doing away with a husband. But she didn't go after her subjects en masse, as did Mary I of England.)

So, although there was a British queen known as "Bloody Mary," no connection between her and the mirror witch has surfaced, save for their both having the same name. Likewise, the "Mary Worth" appellation of the malevolent apparition doesn't appear to be drawn from the lead character of a popular comic strip of the same name. In lore, as elsewhere, coincidences occur. (The "bathing in blood" detail is traditionally attributed to Elizabeth Bathory, but of course her name was not "Mary.")

(Barbara's husband notes: The 'Bloody Mary' legend was common when I was a kid in the early 1970s. We typically performed the "ritual" in bathrooms, because the bathrooms of our suburban homes had large mirrors and were easily darkened even during the day since they had no windows. A familiar 'Bloody Mary' story was one about a girl who supposedly ended her incantation with a spiteful "I don't believe in Mary Worth," then tripped over the doorjamb while exiting the bathroom and broke her hip.)

Why would otherwise rational youngsters want to risk setting a murderous spirit on the rampage? Gail de Vos offers the following explanation:

So why do children continue to summon Bloody Mary, flirting with danger and possible tragedy? The ages between 9 and 12 are labeled "the Robinson age" by psychologists. This is the period when children need to satisfy their craving for excitement by participating in ritual games and playing in the dark. They are constantly looking for a safe way to extract pleasure and release anxiety and fears.

It's possible these "mirror witch" games have their roots in oldtime divining rituals involving unmarried girls and future husbands. There are a number of variations of these divinations, some involving chanting a rhyme in a darkened room on a special night and then quickly looking in the mirror to catch a glimpse of the bridegroom-to-be.

The concept of mirrors as portals between this world and the realm of spirits shows up in other beliefs, namely those surrounding funerals. It was common practice to cover mirrors in a house where a death had occurred until the body was taken for burial. (Back in the days before funeral homes, corpses were washed by the deceased's relatives, dressed in their funeral finery, and laid out in coffins in the front parlor. Consequently, the dead would be in the house for days.) It was believed if the dear departed caught a glimpse of himself in a mirror, his ghost would remain in the house because the mirror would trap his spirit.

Some information from http://www.snopes.com

Other Bloody Mary stories/legends:
http://www.halloween-website.com/bloody_mary_stories.htm