How do you complete the mirelurk mission in fallout 3?

Answer 1

If you are referring to researching Mirelurks in the Wasteland Survival Guide issued by Moira Brown at Craterside Supoply, Megaton, then it can be a very simple quest if done silently.

Only a few steps east of the Anchorage Memorial is a sewer entrance underwater. If you go there instead of the memorial itself (and use a Stealth Boy as soon as you enter), you just creep up to the very first Mirelurk egg sack you see, drop the observation device inside it and then go back to Moira. No harm to the Mirelurks and the main task gets done. You will get the bonus reward.

Come back later if you wish and obliterate anything in the area as there is a unique sledge hammer there to find called The Tenderizer. Good melee damage this early in the game.

Answer 2

Equip a shotgun, go into V.A.T.S., target their face, and press "a."