What's the one, the biggest, life lesson from his grandmother that Nezumi will carry for the rest of his life?
Excellent question! Thank you!
Nezumi's Gran has always been paramount to his development, and her character has influenced Nezumi in every way possible. With the exception of the fantasy verse, where Gran isn't featured, she plays a huge role in Nezumi's upbringing as his primary caretaker. Her approach and her personality shape who Nezumi becomes. From actively spoiling Nezumi and showering him with love, as she does in the 50s con artist verse, to being absent due to declining health, which results in severe neglect in the bar singer verse, and Nezumi falls prey to childhood abuse.
In canon, Gran is an old woman who saves Nezumi from the massacre of his people. She isn't even Nezumi's biological grandmother, but Nezumi calls her Gran and relies on her care from the age of five. She is a bitter and angry woman who lost everything. Her hatred for No.6 is understandable, as it was their army that burned the Mao ( her and Nezumi's tribe ) to the ground, indiscriminately. She has nothing but hatred left in her heart, and little Nezumi feels this.
The lesson that she imprints on Nezumi's young and impressionable mind is as follows: "Never let your guard down. Don't sigh. Don't rely on anybody but yourself. Don't open your heart to anyone. Don't cry." She is part of the reason why Nezumi develops an aversion and fear of human connections. Her influence is crucial, but she is still merely a fragment of Nezumi's trauma, as there is nothing in Nezumi's canon that disproves her teachings, rather the opposite. Nezumi grows up in a world where her lesson is a valuable asset.
Another verse where Nezumi thinks of Gran often, is the theatre verse, where Gran was a bit better role model than canon or bar singer, but she was still very cold and offered Nezumi little to no affection during his development. Her intention came from a good place, but her strict and loveless upbringing left a mark on Nezumi even in adulthood. The words that ring in Nezumi's mind when he thinks of Gran are these: "Always stay strong and independent. Never ask for help, and never lose your face in front of others." And also: "Nobody wants to see your emotions spilt in such an ungraceful manner. Keep them in. They are your own. Not for strangers' eyes to see."
The demon hunter, admittedly, has been the least developed. Nezumi's relationship with Gran was more or less fine. She was a strict woman, but she was also caring, and she showed Nezumi more love when he was a child. The biggest lesson she had given him in this verse was her knowledge of the supernatural. She was the one who grounded him when Nezumi came home after his first accidental kill. She was the one who brought out her books and told Nezumi that she wished she could have protected him from what he saw, but now it's too late. She had told him that because of what he had seen, he was exposed to the supernatural world, and that exposure is like a curse—a curse that would follow him everywhere he went.
"Don't run from it. Accept it for better or worse. Never turn your face away from the truth and live as best as you can," she said. "You're a good boy. You will protect many."
Gran did her best teaching Nezumi what she knew about slaying demons, but she still passed away too soon, when Nezumi was eighteen, leaving him alone to face the harsh truths and follow her teachings.