He knew, you see — he knew your secret — what you did for Dumbledore. And he admired you for it — greatly. And that’s why he named his son — my best friend — after you both. Albus Severus Potter.
Well, here’s CC’s explicit explanation as to why Harry named his son after Severus... he greatly admired Snape’s secret as in what he did for Dumbledore.
It wasn’t because Snape loved Harry’s mother, at least not just that... it’s because of all the heroic things Snape did as "Dumbledore’s man".
It’s also an explaination for the association. Albus and Severus are linked, for Harry, more than anyone else. Without Albus, we couldn’t have won the war. But Albus could never have won the war by himself without Severus, as Scorpius says:
I know without you [Snape] the war could never have been won.
Or, as Scorpius explains more:
I just don’t know what help I — need. Are you still undercover now? Are you still working secretly for Dumbledore?
Dumbledore? Dumbledore’s dead. And my work for him was public — I taught in his school.
No. That’s not all you did. You watched the Death Eaters for him. You advised him. Everyone thought you’d murdered him — but it turned out you’d been supporting him. You saved the world.
[...] Only Dumbledore knew [that you loved Harry’s mother and that you spent years undercover], am I right? And when you lost him you must have felt so alone. I know you’re a good man. Harry Potter told his son you’re a great man.
Naming his son Albus Severus wasn’t done based on a pure emotional whim. Not from Harry, the one who hated Severus with "savage pleasure" (HBP), even when he knew it was unfair and exaggerated.
It was done because Albus and Severus—together—are the reason the world could be saved and Harry could live. Severus wasn’t added just because he loved Lily—he and Albus connected with each other through the war they fought against Voldemort.
Albus entrusted Severus with secrets he could never reveal to anybody (a bit like Harry was regarding the Horcruxes) and which meant he had to remain hated by the world for casting himself as Dumbledore’s murderer instead of his man.
When Snape had to finish Dumbledore off (not even his attempt at healing could get Dumbledore to live much longer), he was—and felt—utterly alone. Because Dumbledore had been his only true confident up until now. He only had a portrait (not the real person) to rely to later... thanks god, or Snape would have to manage the school and the war all by himself otherwise (including, perhaps, as the new unofficial general/leader of the Order). Dumbledore’s death affected Snape more than any other.
I mean, it’s quite something to become the one who finishes your best friend/confident off at their demand, even though you’d never have done it otherwise... putting your own preference aside (of not killing them), risking the integrity of your soul, just to answer the demand of your dying friend, to spare them suffering and give them a clean, painless death at last.
Regulus never did this. Remus didn’t. Cedric didn’t. I doubt they’d ever be able to do this, seeing as what they did and who they became in canon.
The only one who can really connect in the act of inflicting Dumbledore with dark magic is Harry, who gave him poison just to get a Horcrux, following Dumbledore’s previous order despite his pleadings... mere hours before Snape will, at Dumbledore’s demand ("Please"), kill him off.
That’s what the name means, for Harry. He knows both men had their own flaws... he tells Albus Severus this. But he also recongizes it doesn’t make them in less great, and in fact those heavy flaws almost made them greater—as faillible human beings who still strived, together, to save the world he and his family live in.
