Another use for this weapon since it is considered a silenced weapon is for luring. If you have a high sneak (night-time is much easier) and find yourself between two incompatible enemies you can shoot at each of them, draw them toward you while you stay hidden. When they see each other they will attack each other. Hint: Make sure to sneak off to another spot after you lure them or they will see you and a free-for-all will happen.
The version received on your 10th birthday is different than the standard BB gun. It has a higher item health (1,000), a much faster rate of fire, no chance for a critical hit, and does not degrade with use. It is also a Quest item, meaning that it cannot be dropped. There is no way to obtain this version of the weapon outside of Growing Up Fast, except through the use of console commands.
The following is based on Fallout 3 cut content and has not been confirmed by canon sources.
Having limited practical use out in the Wasteland, where BB ammunition is very scarce, this makes the BB gun primarily a sentimental object, as it does less damage than the handgun available from Amata and has a high AP cost.
During the escape from Vault 101, the gun and 50 BB's can be collected from the player's desk. They can offer it to Butch DeLoria to fight off the radroaches attacking his mother (although if his mother dies, he will no longer initiate conversation with the player and the BB gun will be unretrievable without killing him.)
The BB gun is a rare weapon to come across in the Capital Wasteland. With the Point Lookout add-on they are more commonly found, used by swampfolk.