"Harry-Snape is now as personal, if not more so, than Harry-Voldemort."
"‘What would I get if I added powdered root of asphodel to an infusion of wormwood?" (Asphodel is a type of lily and means ‘remembered beyond the tomb’ or ‘my regrets follow you to the grave’ while wormwood is often associated with regret or bitterness)"
The Draught of Living Death"
When I first read the part about Harry naming his Son Albus Severus, I was very puzzled, because I didn´t know what was Harry intention by doing that and to me it seemed like an unnecesary addition by JK to the end of the story, but after some thought I got to some conclusions.
I believe Harry genuinely mourned Snape, the person that gave his life and protected him for years from Voldemort, even if he could care less about the vindictive and petty Potions Master but he also mourned "The Half Blood Prince" that person that helped so much with his potions work, or when Ron got poisoned, who became a kind of friend to him, Harry liked his sense of humor, a friend he felt defensive about, he defended the book from Ginny and Hermione even when he thought about getting rid of it after the Septusempra episode, or even after finding out that Snape was the Prince.
I believe the book was a literary device by Rowling to allow Harry to know Snape as a person, as his mother knew him as a friend or as he used to be, because Snape didn´t allow himself to be that way anymore after becoming a double agent, and specially after Lily´s death, Snape´s life was like a "Living Death" and maybe with the exception of Dumbledore and Minerva, with everyone else he was always minding his reactions, how he could use this or that action, how others would see it, to lie or mislead or to hide his motives. He didn´t allow himself close friendships or love that could be used agaist him WHEN he went back to spy on Voldemort. If Charity is any example, a teacher that used to be his "friend", in her words, we don´t really know how close they were, but that Voldemort still chose to be killed in front of Severus to test his loyalty, he had good reasons to keep himself apart from most people.
I believe Harry understood this after DH, but he still mourned the loss oportunity of having someone share actual memories of his mother with him, not just photopgrahs. Someone who would protect him but also teach him useful skills to protect himself, who understood how he felt about the Dursleys, who knew both sides of the war inside and out, or who would tell him harsh truths even if they hurt, that´s the main reason,imo, why Dumbledore chose Snape to tell Harry he was a horocrux instead of telling him himself.
Harry named his Son Albus Severus as a way to introduce Snape in Harry's family, he understood Severus was "Dumbledore´s man" that he was Dumbledore´s not Voldemort´s, but he wasn´t always that way, he used to be a friend, a Son, a child who dreamed of going to Hogwarts and learn magic, just like Harry himself and Harry wanted to have that small part of him, his best part, that he just found about when it was too late, and heal what was still wrong between them, because Snape died before Harry could tell him all this in person.
I can easily see Harry thinking: "I am naming him Albus Severus POTTER and there´s nothing you can do about it proffesor, that´s what you get for dying before telling me. :)
And we even got Snape´s answer:
“Tell Albus - tell Albus Severus - I'm proud he carries my name.”
Severus Snape - Harry Potter and the Cursed Child