Solomon retained Judah (Jerusalem is the capital of Judah) and was given the one tribe of Benjamin; Rehoboam received the other 10 tribes.
[[ ]] are mine because different names make more sense - to me, anyway.
Benjamin a Light to Judah
What tribe was David or Judah to have? That is going to be Benjamin.
One tribe would go to Judah
“Howbeit I will not take the whole kingdom out of his hand [out of
Rehoboam’s hand][[Solomon's hand]]: but I will make him prince all the
days of his life for David my servant’s sake, whom I chose, because he
[David] kept my commandments and my statutes: But I will take the
kingdom out of his [[David's]] son’s [[Solomon's]] hand, and will give
it unto you, even ten tribes. And unto his [[David's]] son
[[Solomon]] will I give one tribe, that David my servant may have a
light alway before me in Jerusalem, the city which I have chosen me to
put my name there.” 1 Kings 11:34–36
The one tribe to be given to Judah was Benjamin. So God was in this. God was going to give Benjamin to the tribe of Judah.
“And I will take you [Jeroboam], and you shall reign according to all
that your soul desires, and shall be king over Israel. And it shall
be, if you will hearken unto all that I command you, and will walk in
my ways [which Jeroboam did not], and do that is right in my sight, to
keep my statutes and my commandments, as David my servant did; that I
will be with you, and build you a sure house, as I built for David,
and will give Israel unto you. And I will for this afflict the seed of
David, but not for ever [not all the days]. Solomon sought therefore
to kill Jeroboam [Solomon heard about this]. And Jeroboam arose, and
fled into Egypt, unto Shishak king of Egypt, and was in Egypt until
the death of Solomon.” 1 Kings 11:37–40
The prophecy said that Jeroboam was to rule 10 of the tribes. The one
tribe that was to go with David was the tribe of Benjamin. Notice that
back in verse 36, the tribe of Benjamin was to be “a light always
before me in Jerusalem, the city which I have chosen to put my name
there.” Does this mean that the tribe of Benjamin was to be the light
to the House of David? That the tribe of Benjamin had some good
characteristics about it that would help David continually, help the
Jews, and be a light for them and to them? That is a possibility.
In the time of Jehoshaphat years later we read about this light still
being in Jerusalem. Perhaps this light was the tribe of Benjamin. It
seems to read that way.
“And in the fifth year of Joram the son of Ahab king of Israel,
Jehoshaphat being then king of Judah, … he reigned eight years in
Jerusalem. And he walked in the way of the kings of Israel, as did the
house of Ahab: … and he did evil in the sight of YHWH. Yet YHWH
would not destroy Judah for David his servant’s sake, as he promised
him to give him alway a light, and to his children.” 2 Kings 8:16–19