Albert Barnes on Philemon 1:20

Yea, brother, let me have joy of thee in the Lord. "By showing me this favour in receiving, my friend and brother as I request." The phrase, "in the Lord," here seems to mean that, if this request was granted, he would recognise the hand of the Lord in it, and would receive it as a favour from him.

Refresh my bowels in the Lord. The bowels, in the Scriptures, are uniformly spoken of as the seat of the affections--meaning commonly the upper viscera, embracing the heart and the lungs. Isa 16:11. The reason is, that in any deep emotion this part of our frame is peculiarly affected, or we feel it there. Comp. Robinson's Lex. on the word σπλαγχνον. See this illustrated at length in Sir Charles Bell's "Anatomy of Expression," p. 85, seq. Ed. London, 1844. The idea here is, that Paul had such a tender affection for Onesimus as to give him great concern and uneasiness. The word rendered "refresh"--αναπαυσον--means to give rest to, to give repose, to free from sorrow or care; and the sense is, that by receiving Onesimus, Philemon would cause the deep and anxious feelings of Paul to cease, and he would be calm and happy. Comp. Phm 1:7.

(a) "refresh" Phm 1:7