Why Ctrl+P menu would be different, and no "Bone" option On an empty project, when I select mesh and bone, then hit Ctrl+P, I will see "Bone" in the menu.



Download a rig from online
https://dl.dropboxusercontent.com/u/284185/MinecraftPlayer.blend

And try to attach a weapon onto the rig
https://dl.dropboxusercontent.com/u/284185/Bow.fbx

However when try to attach the weapon to bone (extrude right hand with extra bone), menu is now different. Any idea how to have the bone option show up?
That is referred to as a context menu, the available options vary depending on the active item that the action will be performed on. The active object is the one that will become the parent of the remaining selection, it can usually be identified by being highlighted in a lighter colour that the other selected objects.

In your first example you are parenting to an armature.

In the second example you are parenting to a mesh object (that is why you get the vertex parent options).

If you are parenting to a curve object you get curve options.



Parenting to a lattice you get lattice options.



While if you parent to an empty or metaball object you only get simple object parenting options.