Albert Barnes Commentary 1 Kings 8:43

"hear thou in heaven thy dwelling-place, and do according to all that the foreigner calleth to thee for; that all the peoples of the earth may know thy name, to fear thee, as doth thy people Israel, and that they may know that this house which I have built is called by my name." — 1 Kings 8:43 (ASV)

That all people of the earth may know your name, to fear you—Solomon prays that when Yahweh hears the prayers of foreigners directed toward the temple, the result will be the conversion of the entire world to worship Him (Compare to Psalms 96:1-13 and Psalms 98:1-9).

This house... is called by your name—This phrase is meant literally. In Scripture, when God’s name is said to be “called on” people or things, it seems to mean that God is truly present in them, upholding and sanctifying them.

This passage therefore means that when foreigners' prayers directed toward the temple are granted, they will have a full assurance that God is present in the building in a very special way.