In Harry Potter and the Methods of Rationality, we get this quote in Chapter 1:
"Anyway," Petunia said, her voice small, "she gave in. She told me it was dangerous, and I said I didn't care any more, and I drank this potion and I was sick for weeks, but when I got better my skin cleared up and I finally filled out and... I was beautiful, people were nice to me," her voice broke, "and after that I couldn't hate my sister any more, especially when I learned what her magic brought her in the end -"
Later, in chapter 17:
Dumbledore opened the book, seemingly at random, and Harry leaned in to see.
"Do you see these notes," Dumbledore said in a voice so low it was almost a whisper, "written in the margins of the book?"
Harry squinted slightly. The yellowing pages seemed to be describing something called a potion of eagle's splendour, many of the ingredients being items that Harry didn't recognise at all and whose names didn't appear to derive from English. Scrawled in the margin was a handwritten annotation saying, I wonder what would happen if you used Thestral blood here instead of blueberries? and immediately beneath was a reply in different handwriting, You'd get sick for weeks and maybe die.
"I see them," said Harry. "What about them?"
Dumbledore pointed to the second scrawl. "The ones in this handwriting," he said, still in that low voice, "were written by your mother. And the ones in this handwriting," moving his finger to indicate the first scrawl, "were written by me. I would turn myself invisible and sneak into her dorm room while she was sleeping. Lily thought one of her friends was writing them and they had the most amazing fights."
There is further discussion of this in chapter 119, but it's quite vague, and I don't see it as relevant to Lily's motives (although it does shed light on Dumbledore's motives).
What are we meant to infer from this?
The only inference I was able to come up with is that Lily either tried to poison her sister in cold blood, or experimented on her sister (depending on when that note was written). A "potion of eagle's splendo[r]" sounds like it would have worked just fine for Petunia without the substitution. But I'm pretty sure that was not the author's intent.