Fallout – How to rename Dogmeat

When you reach the place marked on your map open the egg clutch-container - it is the nest on the lowest level of the Anchorage Memorial. Then you should browse your inventory and manually place the Observer into the clutch - mission complete.

Maybe the description of "Fallout.wikia" can help you

The easiest and quickest nest to get to without confrontation of a single Mirelurk is the nest located behind the locked [Very Easy] door due southwest from the entrance to Tepid Sewers. After entering, make the first right turn. Directly ahead will be an egg pod. Place the observer and run out.

Stealth Boys are the best choice to reach the optional goal ;)

The basic idea is to move the bobby pin to the correct position then turn the lock with the screwdriver.

The correct position isn't immediately obvious though - you have to test positions with trial and error until you find the right spot. The bobby pin shakes more when it's in an incorrect position. As you move towards the correct position it will shake less. Using the amount of shaking as a measure, you can hone in on the correct position.

Be careful though - when the bobby pin is in a high-shaking position it's easy to break it. If you don't allow the bobby pin to shake too long you should be able to try at least 2 positions without breaking it.

If by the 3rd attempt I don't think I'm getting close, I'll usually exit the lockpicking screen and then try again. This resets the try count (meaning you have more tries before breaking the pin) but also resets the unlock position. If I think I am getting close I might take a chance on a 3rd try. Sometimes it works, sometimes not.

Also, if you are playing on a console, the controller will vibrate when the pin is shaking.

EDIT: There are heaps of how-to videos on YouTube. They manage to get more than 3 tries out of their bobby pins without breaking them... but finer movements look easier on PC with mouse/keyboard controls. (I'm on PS3.)