John Gill Commentary 1 Kings 11:13

"Howbeit I will not rend away all the kingdom; but I will give one tribe to thy son, for David my servant`s sake, and for Jerusalem`s sake which I have chosen." — 1 Kings 11:13 (ASV)

Howbeit, I will not rend away all the kingdom
The whole kingdom of Israel:

but will give one tribe to your son
but it seems he had both Benjamin and Judah, and only ten tribes were rent from him; the reason of this mode of expression may be, either because he gave him one of the tribes of Israel, besides that of Judah, which was his own tribe; or only the tribe of Judah is meant, the whole tribe of Benjamin not being his, since Bethel, and some other places in that tribe, were in the possession of Jeroboam; or rather both these are called but one, because their inheritances lay together, and were mixed with one another; and particularly both had a share in the city of Jerusalem, and the kingdom always after the division went by the name of Judah only: and this tribe was given

for David my servant's sake
because of the promise to him, that there should not want one of his seed to sit on his throne, (1 Kings 9:5)

and for Jerusalem's sake, whom I have chosen
to have the house of his sanctuary and worship in, and therefore thought fit to have one rule there that would have a regard to his service in it.