Ge 19:29 And it came to pass, when God destroyed the cities of the plain (Sodom and Gomorrah included), that God remembered Abraham, and sent Lot out of the midst of the overthrow, when He overthrew the cities in which Lot had dwelt.
When God destroyed and judged Sodom and Gomorrah, he spared Lot for he remembered Abraham who interceded for Lot. Lot was spared for Abraham's sake.
In the same way, when a sinner comes to the Savior for salvation, trusting in Him and what He did on the cross, God saves the sinner for He remembers, as it were, the sacrifice of the Lord Jesus Christ on the cross. He is saved for Christ's sake.
Just as God remembered Abraham so God remembers the Lord Jesus Christ on behalf of the sinner who flees to Him for salvation.