Disclaimer: I own nothing.
Following the terrible and joyous events on Halloween, the Wizarding World scrambled for an explanation for a toddler surviving the most powerful dark lord in living memory. Purebloods paraded the triumph as the Potter ancestry coming through despite Harry's mudblood mother. This was false. Renowned experts in magical theory speculated that Lily and James preformed a variation of the Fraudens Mortem, or cheating death ritual, negating the dark side-effects by using themselves as willing sacrifices. This was also false. The Unspeakables found the evidence contradictory, so the official Ministry report stated the cause as a "magical anomaly", which is the official term for "Merlin knows how". This was closer to the truth. Dumbledore privately thought it was the power of Lily's love, and this was true in a sort of roundabout way.
The truth, though nobody knew, was that sometime between graduating and marrying Lily Evans stumbled upon a blue police box.
The Doctor and Lily, they ran together throughout the whole of time and space. But that's a different story. Our story, the one of Harry's survival, begins at the end of it.
The two landed on the planet Pysphordia and became involved in the villagers' fight against the Euclidiates. The Euclidiates looked like little puppies, with their big, floppy ears and tiny purple bodies. But they were vicious carnivores out of nightmares that fed on the minds of others, using their psychic powers to invade deep inside their victim's consciousness. Lily knew occlumency, that alone was enough to defend herself against them, and in addition the villagers equipped her with a psychic dampening wristband. She was safe. At least, that's what everyone thought.
What actually happened was the wristband dampened her occlumency. She didn't realize because at the moment was a bit busy running for her life. It still would have been fine, except in the process of running for her life a tree branch caught the wristband. In a panic, she pulled as hard as she could. Even then, it still would have been fine, but the villagers had little high-quality supplies due to the war, and the leather strap, already nearly ripped, tore in two. The Euclidiates took her.
The Doctor did not take this news very well. The Euclidiates are extinct now.
Somehow, the Doctor managed to carry Lily back to the TARDIS. She lay too still, but not quite dead. However, with each passing moment more brain cells ruptured. The TARDIS sensed this, and with her knowledge of time itself, knew how to save her. She opened her heart and looked into Lily. A moment later, Lily Evans opened her eyes, the Euclidiates burned out of her head. The next moment, she started to burn too.
Normally, it would be the vortex energy that would kill her, but fortunately vortex energy is close enough to magical energy that her core absorbed enough of it to keep her alive, although she fell unconscious. In this case, the infinite knowledge of time and space contained in the vortex burned her mind. She had to forget.
Fast forward to Halloween night, Lily stood between Voldemort and her only son. Desperately, she searched her mind for, something, anything, to save her Harry. A mother's love is the most powerful force in the universe. So powerful, Lily remembered.
She knew every possibility and might-have-been of the universe. She saw every particle that existed, exists, or will exist. She knew how to create and destroy with only a thought. Not Bad Wolf, that name belonged to another. She was Mother Storm, the one who raged across lands, raining down life-giving water and striking enemies with fire that burned hotter than the sun. Her son received the water, Voldemort the fire. As her last act before falling, she marked her son, her little Harry, with a lightning bolt as a sign to all that the boy was protected.
AN: This is my first ever fanfiction. Critiques are welcome. The background may seem a bit crazy, but considering some of the explanations in Doctor Who it doesn't seem that far-fetched to me. Then again, I'm a bit crazy myself.