2 Samuel chapter 12:
18 On the seventh day the child died. Davids attendants were afraid to tell him that the child was dead, for they thought, While the child was still living, he wouldnt listen to us when we spoke to him. How can we now tell him the child is dead? He may do something desperate.
19 David noticed that his attendants were whispering among themselves, and he realized the child was dead. Is the child dead? he asked.
Yes, they replied, he is dead.
20 Then David got up from the ground. After he had washed, put on lotions and changed his clothes, he went into the house of the LORD and worshiped. Then he went to his own house, and at his request they served him food, and he ate.
21 His attendants asked him, Why are you acting this way? While the child was alive, you fasted and wept, but now that the child is dead, you get up and eat!
22 He answered, While the child was still alive, I fasted and wept. I thought, Who knows? The LORD may be gracious to me and let the child live. 23 But now that he is dead, why should I go on fasting? Can I bring him back again? I will go to him, but he will not return to me.
Notice that last part. David says, "I will go to him." This means that when David dies, he will go to where his dead son is, and we all know that David will go to "heaven" when he dies. This is probably where the doctrine comes from. Sorta tenuous, but fairly clear. The thing about age 12, is probably somebody's guess. If I had to guess, I would say that when a child has their bar-mitzvah he has reached the "age of accountability." I think bar-mitzvah happens on the child's 13th birthday.