How would you write PJO and HOO?
What makes Percy Jackson so interesting-being set in America-is also what makes it problematic. Any excuse given for moving the Greek Gods out of Greece and to America would have been bullshit, but combining ancient mythological creatures and modern day culture and technology can also be a really funny gag (Rick Riordan can write comedy pretty well).
I do think that it could be set in Greece and still be interesting, though, so if I rewrote PJO, I'd do a lot of research and set it in Greece and I wouldn't make fun of the culture. Also I would add the children of Greek immigrants.
I'd definitely get rid of the weird sexism surrounding Aphrodite's daughters and cabin in the books.
I wouldn't make Artemis and her Hunt hate men, I'd make them be clueless about men and turn it into a comedy (men do THIS? No wonder you're so behind! We in the wild do [something completely absurd and simply undoable in modern say society that can be done in her Hunt because it's timeless and in the wild] Far superior).
If I could rewrite HOO, I'd get rid of all the sexism and racism entirely. The characters could actually travel over the world and it would be more impactful to focus on how pollution is affecting the world.
OR, to make it an entertaining and intriguing fictional series-FOCUS ON THE POLITICS. HOO is supposed to be for older audiences, for teenagers, right? Well, plenty of people get into politics at 15-16, so focus on introducing them to politics! See my post here (it's not fully finished but it's better than nothing).