Albert Barnes Commentary 1 Corinthians 7:13

"And the woman that hath an unbelieving husband, and he is content to dwell with her, let her not leave her husband." — 1 Corinthians 7:13 (ASV)

Let her not leave him. A change of phraseology from the last verse, to suit the circumstances. The wife did not have the power to put away her husband and expel him from his own home; but she might think it her duty to be separated from him. The apostle counsels her not to do this; and this advice should still be followed.

She should still love her husband and seek his welfare; she should still be a kind, affectionate, and faithful wife; and all the more so, that she may show him the excellence of religion and win him to love it. She should even bear much, and bear it long; nor should she leave him unless her life is rendered miserable, or in danger; or unless he wholly neglects to make provision for her, and leaves her to suffering, to lack, and to tears.

In such a case, no precept of religion forbids her to return to her father's house, or to seek a place of safety and of comfort. But even then, it is not to be a separation on account of a difference of religious belief, but for brutal treatment. Even then the marriage tie is not dissolved, and neither party is at liberty to marry again.

Regarding "him": the implication is that she is "not to put him away."