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.
In "Escape!" if you attempt to leave the Vault and haven't picked up everything on the desk, a message will pop up saying "You should get your stuff from your desk before you leave" This includes a BB gun. It is classified as being one of the things you must take before you leave and since it is one of your possessions because of the message stating that you must get your stuff before you leave it is the BB gun given to you on your 10th birthday. However, it does not have the unique features like the one given. Your 10th birthday version is taken off you despite being one of your presents and replaced with a normal BB gun that is seen as the one that is given on your 10th birthday. Basically, you retrieve 2 BB guns in the Vault, but one is taken off you and the other is obtained 2 quests later.