Whatever Augustine's questionable understanding of human biology and aging, he clearly understands the point of changing Abraham's name: to foreshadow the blessing that would come through Christ.
When the angels came to Abraham and Lot (promising blessing and judgement respectively), they both recognized the hand (and possibly even the person) of God in the visit.
In Lot's escape and Abraham's sin (lying to the king about Sarah), we see a warning for believers. A delicious warning:
For what is meant by the angels forbidding those who were delivered to look back, but that we are not to look back in heart to the old life which, being regenerated through grace, we have put off, if we think to escape the last judgment? Lot’s wife, indeed, when she looked back, remained, and, being turned into salt, furnished to believing men a condiment by which to savor somewhat the warning to be drawn from that example.