There is always a period between clinical death, when the heart stops, and true death. The hospital doesn't perform any resuscitation on a patient known to be dying of cancer, but Earth is not particularly technologically advanced.
In the moment where Meredith is dead but not dead, a small alien implant in her body kicks into gear, essentially rebooting her. Her heart stutters and resumes beating; her eyes flutter open; in the tests that follow, there is no sign of cancer in her body, nor the remaining ravages of the chemotherapy and radiation.
By then, Peter is gone.
Peter's Walkman is twenty years old, patched to hell and back with a mismatch of various sorts of alien tech to keep it running. But it works.
He tries not to sing along too loudly when he's not alone, but the bar's noisy anyway.
"So I waited with high hopes / and she walked in the place / I knew her smile in an instant / I knew the curve of her face / It was my own--"
The music goes on, but he stops dead, staring, unable even to breathe. It's impossible, it's impossible, but -- "Mom?"
She nods, holds out her arms, and Peter runs.
Catching up takes a while, and Peter goes through a riot of emotions that ends on giddy elation. His mom is alive, and she's found him, and nothing is going to separate them ever again.
He laughs, and twirls her in a dance move, and starts singing -- only to stop cold when she frowns and says "What is that noise?"
With hands that start to shake, he puts the Walkman headphones to her ears and presses play.
"Ugh," she says, shuddering.
That night, alone again, Peter stares at the cassette tape for a long moment before smashing it to pieces.