When Harry was younger, he used to draw on his cupboard with whatever he could, at first crayons and then pencils and markers. As he got older, he got better, drawing more complex and intricate designs. Through primary school, he drew flowers, trees, anything that caught his eye, anything that he thought was beautiful.
In his first year, he drew Hermione and Ron in the great hall, in his second year he drew the Basilisk, and in his third, he drew his mother and father, and in his fourth year, he drew the Hungarian Horntail, using pencil and coloured pencils. He drew on the walls as an escape, but maybe it was more than that.
Maybe the whole point of it all was for him to say that he was here, that he was alive, that he existed in the grand scheme of everything. That he was something. That he did something, even if it wasn't much. He thought he was nothing and so he made himself something. Not that anyone would have known. Who would he tell? Not his teachers, not his peers, not even Hermione and Ron.
Maybe the point was purely a reminder for himself. That yes, he exists, he does matter. Despite what he's been told by the Dursley's and everyone else, he could know that he exists, he matters. Because he does, why else would he be here? Why else would he have survived? Why else would be have lived when he could have died so many times?
He had to matter. He had to survive. He was here. He did exist. And there had to be a reason.