Beast: While it appears that Hank was born with his large hands and feet, it is highly likely that his secondary mutation of blue fur, claws, and fangs was a direct result of his having been called "Beast" for so long. He was a Beast in name, and then he became a Beast in fact.

Cannonball: Sam's power manifested during a mine cave-in. He was trying to save an injured man, whom he was carrying in his arms. At that moment, Sam needed to get out of the mine, without getting hurt, and he couldn't use his hands. Hence, a thermal invulnerable blast field that emanates from his legs and allows him to fly.
Sources: Marvel Graphic Novel #4: The New Mutants

Colossus: Assuming that the scene in Giant Size #1 is the first time that Piotr used his power, it is clear that he needed to become stronger and tougher in order to save his sister from an oncoming tractor. What better way to stop two tons of steel than to become steel yourself?
Sources: Giant-Sized X-Men #1

Cyclops: Scott's first use of his power was while falling out of his parent's airplane. He was holding on to his brother Alex and looking straight at the ground. The parachute was failing, and Scott couldn't do anything but watch, so his eyes began to emit beams of force that pulverized the ground beneath them and allowed them to survive the landing. While it has been stated that Cyke can't control the beams because of brain damage caused by the original fall and Sinister's experiments while he was in the orphanage, it is also possible that Scott has a psychological hang-up as well. He can't stop the beams because somewhere in his subconscious, he's still falling out of the plane, and his power is the only thing that can save him. To shut it off defies self-preservation. This feeling could have caused or exacerbated the brain damage Scott suffered in the fall from the plane, kind of like a power overload that blew out the circuits.

Dazzler: Alison Blaire always had strong ambitions to be a singer. Her father, however, always condemned it, and tried to shut her off from her dream. As Alison's body matured, she was able to transform sound, even her father's voice, into dazzling light which would "leak" from her body, making herself (literally) able to shine like a star.
Thanks to Havoc1 for this info.

Emma Frost: While her initial power of telepathy may or may not have had its roots in Emma's adolescent home life, it is clear that her secondary mutation of generating a hard diamond skin was a direct response to the Sentinel attack on Genosha that killed her students and buried her in tons of rubble. It is also likely that the psychic deathscreams of 16 million mutants so traumatized Emma that she cannot use her telepathy in the diamond form, because she subconsciously repressed that power the first time she became diamond-hard. The diamond form also matches the psychological makeup of a cold, emotionless person that Emma projects to keep herself from feeling the loss of her second group of students.
Sources: New X-Men #116, New X-Men Annual 2001

Greystone: Greystone manifested during the "M" branding process in the mutant camps in Bishop's future. He needed size and strength quickly. His grotesqueness may be a side-effect of his revulsion due to the branding and scarring.
Sources: X-Factor #145

Havok: Alex wanted to hurt a bully who had been beating up on him the night he got his powers. He absorbed energy from the stars, the only significant source available at night, and channeled it through his fists.
Sources: X-Factor #-1

Husk: Paige always wanted to be someone other than who she is, so that she could be an X-Man, like her brother. That's why she studies so hard, and why she tries to cover her Midwestern accent. Her powers, therefore, reflect this desire, and allow her to literally shed her skin, and become something, if not someone, else. In fact, the night she manifested, she yelled up to God on frustration, "If y'all don't make me a mutant, ah'm gonna tear right out of mah skin!" and that's exactly what she did.

Leech: Growing up in the Morlock tunnels, Leech was under the constant threat of older, bigger mutants, most of whom had powers that could do him serious damage. In this stressful environment, Leech's body and mind instinctively locked on to the ability to drain powers, to be used as a defense mechanism. When the others couldn't use their powers against him, Leech became relatively safe.

Magik: As a captive in Limbo, Illyana needed to escape, so her powers locked on to Limbo's stepping disks and allowed her to control them.
Sources: Magik LS #1-4

Magma: Amara was sacrificed to Selene by being thrown into a lava pit. Rather than surrender to death, her mutant powers manifested (under a great deal of stress), and she merged with the lava, using her energy to manipulate and control it.
Sources: New Mutants #10

Magneto: During the Holocaust, the young Magnus was lined up with his family at a mass grave. The German soldiers then shot them all. Magnus manifested his power under the stress and managed to deflect the bullets away from himself, though he fell in and was buried anyway. He later crawled out and was returned to the death camps. The trauma of the camps and the malnutrition he suffered in them delayed his full powers from emerging for years, but when they did (against the crowd who would not save his daughter Anya from a burning house) his immediate threats were again made of metal.
Sources: New Mutants #49, Magneto #0

Multiple Man: Madrox grew up on a Kansas farm alone, as his parents died when he was very young. When his mutant powers kicked in, his mind directed them to create companionship for him. He's his own best friend.

Nightcrawler: Kurt's physical mutations were apparent at birth, but his signature brimstone "BAMF" may have resulted from his feeling like a devil, due to his appearance.

Phoenix: Jean's powers first appeared when her friend Annie Richardson was hit by a car. In that intense moment of grief and stress, all Jean wanted in the world was for her friend to hear her and wake up. Her body obliged by developing telepathy. Later, when her TP was blocked out, it rerouted the energy into telekinesis, which appears to be governed by the same area of the brain.

Professor X: Charles first used his power to soothe his mother's mind after being abused by Kurt Marko. He would have wanted her to hear him, and she probably wasn't paying attention, so his mind developed telepathy to help her calm down.
Sources: Uncanny X-Men #309

Rachel Summers: Rachel inherited her TP and TK from her mother, but her own power is to psychically shift herself in time. This fits with her desires, as she always wanted to change the past, and run away from her present.

Rogue: Rogue first used her power during her first kiss. Every girl wants to know what the guy she's kissing is thinking, so her powers became a form of telepathy, that absorbed the memories and thoughts of whomever she was touching. The effort was so great, however, that she had to drain his energy, and the horrifying shock of that may be what caused her power to be out of control.

Rusty: Rusty was being "propositioned" by a woman when his powers emerged. The physical and hormonal stress of the event caused his temperature to rise and triggered his latent mutant gene. However, he got so hot that his body needed to adjust to control the heat, resulting in his powers as a pyrokinetic.
Sources: X-Factor #1
Thanks to Sy S. for this info.

Sauron: While Sauron has been called a "non-mutant variant," it is possible that he is really a mutant, or that the theory applied to him anyway (see Earth X up top). The pterodactyl attack that Lykos survived as a boy made him very sick, and was terribly stressful. The shock and trauma of the ordeal activated his latent powers, which sensed that his body needed more energy if it was to survive. He developed hypnosis to lure his dog to him (it probably didn't want to come), and then his powers latched on to energy-drainage, to provide his body with more energy, allowing him to heal and survive. The secondary mutation into Sauron occurred because of the large amount of energy he absorbed from Havok. Again, the stress of the event activated the mutant gene, and with it, a physical change. This time, he turned into a humanoid pterodactyl because the original attack was on his mind, having just recently resurfaced as memory after being supressed for so many years.
Sources: (Uncanny) X-Men #60

Shadowcat: At the time of manifestation, Kitty's parents were splitting up, she was having intense headaches, and she had just been introduced to Emma Frost, who really freaked her out. All Kitty wanted was not to be hurt, and not to be touched or taken away by Emma, or anyone else, for that matter. As a defense mechanism, then, her body directed its mutancy to turn Kitty insubstantial, so that no one could touch her or hurt her again.
Sources: Uncanny X-Men #129

Skids: Skids was abused by her father when she was a child. Naturally, her mutant powers evolved to allow her to protect herself - with a slippery impenetrable forcefield.
Thanks to Jarrett Anzalone for this info.

Storm - Apparently, the first time Storm used her powers was to help her village during a drought.

Strong Guy - Like Sunspot, Guido manifested during a fight. The difference is that Guido was getting beaten up badly. He needed strength and size, but his body took the energy to do it from the punches he was getting (kinetic energy). He was also something of a runt, and wanted to be bigger and stronger even before the fight.
Sources: X-Factor #87

Sunspot: Bobby was in a fight on a soccer field when his powers bloomed. He needed strength and invulnerability at that moment, and to get it he absorbed the most readily available energy: sunlight. The emotional stress and anger over being racially insulted because of his skin color may have also contributed to his pitch black appearance.
Sources: Marvel Graphic Novel #4: The New Mutants

Thunderbird: John Proudstar wanted to be the ultimate Apache: fast, strong and sharp. Assuming this was a childhood dream, his powers appeared to have complied with his state of mind.
Sources: Giant-Sized X-Men #1

Warpath: James Proudstar always wanted to be exactly like his brother. When his powers blossomed, that's exactly what he got.

Wolfsbane: If we assume that Reverend Craig was calling Rahne "devilspawn" before her mutant powers arrived, because of her red hair and her (possible) illegitimacy (she might actually be his daughter, and he projected his sin onto her), then it is understandable how her mental state at the time influenced her mutancy to turn her into a wolf, one of the Devil's forms, and also into a transitional "werewolf" state, which is also a type of "devilspawn."
Sources: Marvel Graphic Novel #4: The New Mutants