The Duty of One (Mother) becomes The Duties of Two (MIB and Jacob) by BrewsterDMB

Here's my guess at what played out, assuming that the Source is actually a "religious" entity, and not a scientific one that is just deemed religious because of ignorance and failures of the people of the time to understand electromagnetism and science.

Mother was given two sacred duties:
1. Keep mankind from getting into the Source (Claudia's people, Dharma), and
2. Keep those who have moved on from coming out of the Source (ghosts, whispers, etc)

Going from the aftermath of what happened after she knocked MIB's head into the wall, I'm going with Vozzek69's claim that she was A smoke monster; I say "A" instead of "THE" because I don't believe she was the same as the one that MIB became.
This theory spawned from his, so I suggest reading his first (which is totally more coherent).

Feeling that her replacement had come to the island inside Claudia, by her own doing most likely (which is a Jacob quality), she was preparing to instill these duties into Claudia's son.
However, she found herself caught off guard when Claudia gave birth to two sons.
So instead of being given one person completely capable of fulfilling duties 1 and 2, she is given Yin and Yang.

Jacob grows and becomes attached to the island, determined to protect the Source from being acquired/reached by man (I believe he ordered the Purge of Dharma because they were getting too close to abusing its power). Maybe the Source lets man live forever? Man HAS to die. It is not up to Man to manipulate "passing on", whether it be to heaven or hell.
So it is my theory that he has been granted with Duty 1.

MIB has always wanted to leave the island, and who can blame him?? His mother lied to them both, whether it was self-serving, or really for the good of the island, or for the brothers, is another discussion. But MIB can see the dead, and Jacob cannot. He is more closely connected to death. Note: MIB has been referenced by Dharma as Cerberus, which "guards the gates of Hades, to prevent those who have crossed the river Styx from ever escaping." The Boy in Black says that it is beautiful, while Jacob seems indifferent.
So it is my theory that he has been granted with Duty 2.
(The only outstanding question is why the MIB is unable to find the Source for 30 years; maybe because he relinquished himself to living with Mankind, and therefore gave up the right to be able to see it?)

I'm not sure how this translates to the fact that Jacob has a name but MIB doesn't. Names are earthly labels attached to a body, while in the afterlife all we have our souls, stripped of any earthly attributes, and therefore have no need for one?

Aside: MIB's mother calling him to go back to the others is very reminiscent of Ben's mother calling him. In the end, Ben is the one who "frees" Jacob from his duties, in the same way that MIB frees his Mother from hers. Does this mean Ben will the new protector?