ALBUS: Slytherin is the House of the snake, of Dark Magic… It's not a house of brave wizards. (1.2.36)
Does Albus reform Slytherin's reputation, or does he confirm it with his actions during the story?
GINNY: People always look when you three are together. And apart. People always look at you. (1.2.65)
This line confirms that Harry's still a celebrity in the wizarding world. It's impossible for him to keep a low profile…and that probably contributes to him being a wee bit self-centered.
ROSE: I'm a Granger-Weasley, you're a Potter—everyone will want to be friends with us, we've got the pick of anyone we want. […] We rate them all and then we make a decision. (1.3.10)
How did Ron and Hermione end up with a popularity crazed daughter like Rose? She has Hermione's intelligence, but she applies it in a cold way to acquire a squad.
ROSE: Your mum and dad are Death Eaters! (1.3.33)
Rose thinks she should be famous because of her parents. Along those same lines, she views Scorpius as infamous because of his father, Draco Malfoy.
DRACO: Harry just wants his face back in the newspapers again. (1.12.12)
Draco believes that everything Harry does is motivated by fame. Do think there could be a kernel of truth to it?
AMOS: So I should allow you to get involved because you're a Potter? Relying on your famous name, are you? (1.14.8)
Just a couple scenes later, we see someone accusing Albus of capitalizing off Harry's fame. This is a bit hypocritical of Amos because he wanted famous Harry Potter's help, yet accuses his son of trying piggyback on his famous dad. Fame stinks.
SCORPIUS (exploding): Try my life! People look at you because your dad's the famous Harry Potter, savior of the wizarding world. People look at me because they think my dad is Voldemort. Voldemort. (2.16.48)
Albus is famous, but Scorpius is infamous. Which is worse?
SCORPIUS: You're Head of Magical Law Enforcement? (3.3.5)
In bizarro world, Draco has the same job Harry has in the original timeline. Draco abuses the power and prestige that Harry has, using it torture Muggles. Harry may kind of stink as a dad, but at least isn't a terrible person.
CRAIG BOWKER JR.: You're the Scorpion King. Of course you hate homework. (3.4.10)
Scorpius has a great reputation in the bizarro world. He's like the Anti-Potter, just as popular as Harry Potter, but for completely different reasons. Although Harry had his detractors (like Draco Malfoy), bizarro Scorpius eliminates them with the threat of torture.
ALBUS: If it wasn't for you, everything could have gone Dark. (3.14.24)
Harry won't do anything to help Scorpius (and Draco's) reputation, but Albus appreciates Scorpius for who he is on the inside. And Scorpius learns to be gregarious and outgoing and reform his own reputation.