Adam Clarke on Genesis 16:14

Wherefore the well was called Beer-lahai-roi - It appears, from Gen 16:7, that Hagar had sat down by a fountain or well of water in the wilderness of Shur, at which the Angel of the Lord found her; and, to commemorate the wonderful discovery which God had made of himself, she called the name of the well באר לחי ראי beer-lachai-roi, "A well to the Living One who seeth me." Two things seem implied here:

1. A dedication of the well to Him who had appeared to her; and,

2. Faith in the promise: for he who is the Living One, existing in all generations, must have it ever in his power to accomplish promises which are to be fulfilled through the whole lapse of time.