My friend, who has naturally dark brown eyes, wants to change her eye
For what reason? If it was other than treating an illness or for fixing a deformation then it is considered Haram (prohibited).
The Prophet (Peace Be Upon Him) said:
"Allah has cursed the females who get a tattoo and the females who tattoo, the females that pluck their eye brows and the females who pluck them, the females who file their teeth for beauty and the females who change the creation of Allah."
(Sahih Al-Bukhari and Muslim)
She doesn't want to get surgery. She's considering using eye drops that with regular use lessen the amount of melanin in eyes
Changing Allah creation of you by getting under the knife, by using drugs (as in your friend case), or by natural remedies, are all unacceptable ways and can't justify your action/doing as they result in the same outcome which is the issue here.
and permanently change their color.
If it was a temporary change with staying around women and/or Maharam, it would be okay (such as using makeup). But it is permanent, and she would get out with that change in her face, which change nothing in it being prohibited in Islam.
Question: As women are lightening the melanin in their skin and hair
and that's not considered wrong, if my friend does the same to her
eyes, would this be wrong?
Who said that those are not considered wrong?