Memory Alpha is a system in the Memory Alpha block just Galactic North of SOL.
The Quest to deposit Egg's sample ends there with Commander Romaine. Cmdr Romaine is in the center of the complex on the lower floor.
Once you speak to her you can turn Egg's sample in at the "mailbox" interfaces in front of her ( and by in front I mean she's facing them ).
You can also start turning in ore, alien, energy, and all of those other miscellaneous items that you've been picking up that no-one else wants to buy. You turn them in for items and upgrades here. Be warned it takes a LOT of stuff to get anything.