Albert Barnes Commentary 1 Kings 8:41

"Moreover concerning the foreigner, that is not of thy people Israel, when he shall come out of a far country for thy name`s sake" — 1 Kings 8:41 (ASV)

Nothing is more remarkable in the Mosaic Law than its liberality with regard to strangers, both in general (Exodus 22:21; Leviticus 25:35; Deuteronomy 10:19) and in religious matters (Numbers 15:14–16; Deuteronomy 31:12). It is quite in the spirit of these enactments that Solomon, after first praying to God on behalf of his fellow countrymen, then intercedes for the strangers, asking for their prayers the same acceptance he had previously begged for the prayers of faithful Israelites.

For your name’s sake—that is, "to visit the place where you have set your name" (Compare to Deuteronomy 12:5, 11, etc.).