Here is a family tree theory I came up with. It shows a way that Harry could be related to the Weasley family by Molly Weasley (née Prewett, who in this theory would be his third cousin once removed) marrying into it.

First off, this would mean that Molly Weasley's family, the Prewetts, were not entirely pure-blood, with a few Muggles having married into the family. It would also mean that Lily Potter (née Evans) simply thought she was Muggle-born, not knowing that her father--also Molly Weasley's second cousin, who Ron mentioned was an accountant--was a Squib. (This is entirely plausible; the Wizarding world must keep its existence a secret from Muggles to the fullest extent possible. Also, as revealed by J.K. Rowling herself in a post-Deathly Hallows interview, Muggle-born wizards and witches would not have magical powers if they did not have a Squib somewhere on their family tree.)

More significantly, it would also mean that Harry and Ginny were distantly related--but is their relation distant enough that this wouldn't get in the way of their relationship?

The Weasley family would also not be related to the Potter family in this theory until Lily (a fairly distant relative of theirs) married into the Potter family; this would explain the difference in financial situations between the two families.

If this family tree theory is true, then one might wonder why Harry went to live with the Dursleys, who treated him far worse than the Weasleys did when Harry was at the Burrow. The magic behind Harry living with his mother's blood would still be effective if Molly Weasley and Lily Potter were second cousins, as they are in this theory. Therefore, the only reason Harry would continue to live with the Dursleys despite their mistreatment of him, is if there were a legal requirement that Harry had to live with his closest living relatives--who would still be the Dursleys.

Here's another theory of mine: What if the entire pure-blood supremacist ideology is nothing more than the Wizarding world's revenge on the Muggles for forcing the Wizarding community into hiding? Just so you know, in no way do I believe that this justifies what is basically the Wizarding equivalent of racism.