Notice! This mission will be available only after you complete Path of Knowledge!
Neloth needs a very untypical ingredient for his experiments - the Briarheart. The item can be dropped only by Forsworn Briarhearts, who replace their organs with them. There aren't many of them in Skyrim, but you can find a few in locations such as Bard's Leap Summit, Lost Valley Redoubt or Sundered Towers.
Note that you don't even have to kill a Forsworn Briarhearts - just steal from him. If you take away his most vital organ, he will immediately die without making any noise.
Talk to Neloth
After completing the mission, return to the mage of Tel Mithryn. He will take out memories out of the Briarheart and reward you with a given (not very high) amount of gold. Don't worry, as completing mission for Neloth is profitable in long term - it will let you unlock some more advanced (and lucrative) quests.