mass-effect-3
I picked up that artifact on a planet somewhere. Now there's a mission in my journal saying that I should find someone on the citadel who can use it. I've been running around the place a long time, but I can't seem to find anybody.
Who wants the banner?
A human doctor will be looking for the medi-gel formula you have in your possession following the completion of your mission on Palaven. You can find her in lobby of the Huerta Memorial Hospital on the Citadel, to your left when you walk in the door:
Outside of having completed the Palaven mission, there's nothing in particular that you need to do to trigger this mission beyond talking to her once in possession of the forumla. As Raven Dreamer notes in the comments, the doctor does have a side-conversation that would clue you into her need for the formula to begin with which you'll overheard as you approach her.
No, the journal gives no indication whether you've completed the objective or not. If you have, you will be able to talk to the quest giver, otherwise you won't.
Best Answer
There is a group of Turians in the Purgatory bar that will accept the banner. Doing so will earn you the war asset of the Turian 7th fleet.