"So young black women can only look up to/be inspired are people that superficially look like them??"
You're putting words in my mouth here. I specifically said: "Having a Black Annabeth is going to be so inspiring to so many young Black girls out there and that, in my opinion, can ony be a positive thing." I never said that you can only be inspired by someone of the same race as you. I don't agree with that at all. You can be inspired by anyone of any background. But when you have been deprived of on-screen representation your entire life, having someone in a TV show who looks like you means more than you know.
Forgive me if I'm wrong, but I believe you're caucasion? Western media has always been dominated by white people, so you probably can't relate to what it's like to not have racial representation. For me, growing up as Asian in the UK, I rarely saw Asians on screen. (Take your example of the Hunger Games - can you name me a single Asian character in the whole series? Because I can't.) If by some small chance there were Asians, they would usually be the stereotypical "ugly" nerdy side characters and the butt of the joke. They would rarely be fully developed, three-dimensional characters who I could look up to. The same goes for pretty much every ethnic minority out there. Imagine seeing story after story in the media where the one token person of your race is pushed to the sidelines and is nothing more than a raging stereotype. You see it enough times, and you start to believe that's all you're going to amount to.
This quote from Awkwafina hits the nail on the head for me: "That is the power of representation: you don't realize how important it is until you realize you have been missing it your whole life"
Can you think of a story featuring a strong, brave and clever young heroine who's white? I don't need to look any further than your post to find an example: The Hunger Games. Now, can you think of a story featuring a strong, brave and clever yoing heroine who's Black? Harder, right? Most of the examples will be stories specifically written for Black characters, like The Hate U Give. But the majority of stories which don't centre on race won't bother with a poc main character.
This is why it's so important and so meaningful that Rick Riordan's cast two actors of colour in the PJO trio.
I think Hippophile sums it up really well:
But to me, the whole having an issue with the casting of Annabeth as black is a little ridiculous. It doesn't change her storyline, like you said, and it's always nice to be more inclusive to other races, imo. In a world with a lot of TV being dominated by a largely white cast of characters, if not all white, I have no problem seeing characters being taken on by actors/actresses who can do it justice.
I find it interesting you used Cinna as an example, because he was never specified as Black in the books. When a Black actor was cast, many (racist) people were unhappy about it. But the actor did a fantastic performance and, as you said yourself, it's now hard to imagine Cinna as anything other than a Black. This only goes to solidify my point that Black actors can 100% do roles justice that weren't originally written for them.
You don't create diversity by stealing other characters, you do it by creating your own.
Rick Riordan created the character of Annabeth so I don't see how he could have "stolen" it?
"and lets say for argument's sake that this TV show inspires this young black girl to read the source material. If her only connection to the character is the fact that she's also a young black girl, she's going to be extremely disappointed. That no, in fact, the character wasn't even black and someone basically hijacked the character. How do you think she's going to feel?"
- Let me explain why I disagree with this. I watched Shadow and Bone, a TV series with a half-Asian actress playing the main character, even though she was described as white in the books. I then read the books, and continued picturing her as half-Asian, because that's the beauty of books - so much of it is left to your imagination. I didn't feel like the books had been hijacked. If anything, I felt grateful that the casting director had taken a chance on an amazing Asian actress who truly enhanced the character and gave her a lot more depth (as the racial aspect added more layer to her backstory of being shunned and feeling like an outsider), which carried over to my reading experience.
You're right that Rick's later books (Heroes of Olympus onwards) have great character diversity, but this TV show is an adaptation of the initial Percy Jackson series, where the characters are predominantly white. So it makes sense Rick would want to add more diversity for this adaptation.
"Annabeth always inspired me and that isn't because she is a white skinned, blonde hair girl. It was because I was inspired by her quick thinking and her relationship with others. That isn't based on the fact that she is black or white."
- ok, so if you are inspired by Annabeth regardless of her appearance, it shouldn't be too much of an issue that she's going to have brown hair and dark skin instead of blonde hair and light skin, right?
" If I was a little black girl and I watched the show I would probably feel like they changed her skin tone to make her more relatable to me, like I'm not smart, cute, funny and strong, like they had to change her to make her relatable to me. if she was white and I wasn't I would feel like, hey, this girl isn't the same color as me but I relate to her personality, which is the only thing that matters!"
- This is funny, because it's exactly what people of colour have had to do their entire lives. We've had to find ways to relate to characters with completely different cultural backgrounds and appearances as us, because we have no other option. We don't see ourselves depicted in the media nearly as much as white people. And why might you think a Black girl would watch the PJO series and think they cast a Black Annabeth purely because of diversity points and not because a Black girl could actually be smart, funny and strong? Because society has led her to think that way. It's led her to believe that a heroine like Annabeth could never be Black just because.
"along with what whats-her-face said" - if you want to have a respectful discussion, I'd appreciate if you could take the 2 seconds to scroll up and see what my usename is rather than calling me "whats-her-face" 
"And percy, it's just hair! Maybe they'll dye it or have a wig. If they don't. I'm fine with it, he's a really good actor and I think he can really sell me on percy. But I haven't seen any movies with Leah so I don't know her acting style."
- So, if I'm hearing you correctly, you are fine with Percy's appearance is different because you've seen the actor's work already and you know he can pull it off. But you haven't seen any of Leah's work yet, so you are judging her ability to play Annabeth on no basis except her appearance?
Honestly, I'm tired of seeing backlash against Leah Jeffries. She's literally a 12 year old and doesn't deserve any of it.
Anyway, sorry if any of this reads as argumentative, that wasn't my intention. I just want to get my point across
