We observe that Petunia says took a potion that made her sick for weeks then she became beautiful. (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 -"

We observed that it Harry thinks her beauty is a sign of magic and deduced that it was extremely rare and dangerous. (Chapter 36)

"Harry?" called a thin, blonde woman whose perfectly smooth and unblemished skin made her look a good deal younger than thirty-three; and Harry realized with a start that it was magic, he hadn't known the signs before but he could see them now. And whatever sort of potion lasted that long, it must have been terribly dangerous, because most witches didn't do that to themselves, they weren't that desperate...

We observe that Dumbledore admits to writing into Lily's potion notes. Specifically he claims he wrote notes regarding a potion of Eagle's Splendor that includes a modification where you get sick for weeks and is very dangerous. (Chapter 17)

"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 splendor, many of the ingredients being items that Harry didn't recognize 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."

We observe that Harry and Eliezer are familiar with Advanced Dungeons and Dragons (chapter 3); although Harry may not be familiar with this particular potion (D&D 3.0 came out in 2000). Eliezer uses the words "Eagle's Splendor" when that already is a well defined term in D&D. (From the D&D SRD)

Eagle's Splendor

The transmuted creature becomes more poised, articulate, and personally forceful. The spell grants a +4 enhancement bonus to Charisma, adding the usual benefits to Charisma-based skill checks and other uses of the Charisma modifier. Sorcerers and bards (and other spellcasters who rely on Charisma) affected by this spell do not gain any additional bonus spells for the increased Charisma, but the save DCs for spells they cast while under this spell’s effect do increase.

Now, it is possible that Petunia and/or Dumbledore are both lying and/or false memory charmed. We do only observe them saying things and not the actions themselves. It could also be coincidence that Eliezer used "Eagle's Splendor" as a borrowed keyword or he may be trying to mislead us. But these all require an unnecessary addition to our theory.

I suspect we're supposed to go with the obvious answer. The answer that these people are not lying or memory charmed: Lily gave an unusual beauty potion to Petunia; Dumbledore wrote in the margins about a modification to a beauty potion; the side effects for both unusual potions were the same. Ergo Dumbledore, as Lily's potion confidants, helped her come up with the components of Lily's unusual beauty potion.

If you have other observations or come to a different conclusion, feel free to share it. But absent of further evidence, I consider it obvious that Dumbledore helped Lily devise the potion and that Eliezer waned this to be obvious to us, rather than a mysterious secret. Do you come to a different, less wild hypothesis, given these observations?