Genesis 21:1

And the Lord remembered Sarah according to that which He had said to her; and the Lord wrought a miracle for Sarah like to that for which Abraham had spoken in prayer for Abimelek.

AND THE ETERNAL ‘PAKAD’ (VISITED) SARAH AS HE HAD SAID. That is, by granting her pregnancy. And the Eternal did unto Sarah as He had spoken by granting her the birth of a son..Thus the words of Rashi.
But the word pekidah is only an expression of remembrance and attention to the one who is remembered, such as: G-d will surely remember (‘pakod yiphkod’) you; I have surely remembered you; And Samson remembered his wife by [bringing her] a kid. Here, too, the sense of the verse is that the Eternal remembered Sarah, and He did to her as He had spoken. This expression is also found in connection with all barren women who later give birth. Thus, in the case of Rachel: And G-d remembered Rachel; and in the case of Hannah: And G-d remembered her. Similarly, the Rabb is said, “Biblical verses which mention pikdonoth are equivalent to verses which mention Divine remembrances.” is treated as one mentioning remembrance. Thus, Ramban proves that the word pakad in the verse here can mean “remembered,” and not as Rashi explained it as meaning “granting pregnancy.”

וה' פקד, whereas Avraham was busy praying for the entourage of Avimelech to be able to give birth to the various fetuses all of which had had their progress towards birth arrested when Avimelech had taken Sarah into his palace, G’d used this time to make Sarah conceive. We find that something similar occurred in Job 42,10 on the occasion when Job is portrayed as offering a prayer on behalf of his friends Eliphaz, Bildad and Tzofar, who had been afflicted by G’d for having criticised G’d’s justice just as had Job at the time. (compare Rashi).

כאשר אמר, a reference to 17,16 where G’d told Avraham that He would bless Sarah (as distinct from Sarai). The blessing consisted in that G’d removed the curse from which other women suffer in that both their pregnancy, giving birth, as well as raining their children are frequently very frustrating experiences as we know from Chavah in Genesis 3,16 that the whole process of pregnancy and giving birth would be an extremely painful experience.

ויעש ה' לשרה כאשר דבר, in accordance with the promise in 17,16 “I will also give you a son from her.” When we do observe old men becoming fathers, the child is almost always a female. G’d had promised and kept His promise that the child Sarah would bear for him would be male.

וה' פקד את שרה, she conceived and gave birth in accordance with the times predicted. Seeing the matter was something so unusual, miraculous, the Torah repeats this fact repeatedly. Normally, when the Torah repeats certain information it does so by using different words, changing the syntax; Here too, the verse commences with the verb פקד, but then uses the word ויעש to tell us the same thing. At the very end of the verse it changes the syntax a third time, writing כאשר דבר, “as He had said.”