If we take a step back and look at the developments in Bereshis beginning with Yitzchak's birth, we can detect a sublime theme. The name Yitzchak - as is clear in many passages which relate to his birth - is due to the tzchok - the laughter - which was precipitated by his birth. Laughter represents that which is unreal, which does not coincide with the world as we know it. Whether laughter is a response to extreme simcha (see Rashi based on the Midrash regarding the many joyous events which transpired on Yitzchak's birthday) or an expression of disbelief, it connotes that which is beyond reality. Yitzchak's name, moreover, indicates that his persona was other-worldly (see Ramban in Toldos), and - as we know from the Midrash (quoted by Rashi) which describes Rivka's reaction to her first sighting of Yitzchak - he had an angelic, holy aura about him.
In light of the above, we can better understand the import of God granting Yitzchak to Avrohom and Sara as it pertains to Rosh Hashanah. A significant portion of our prayers to evoke God's mercy is in the framework of His remembering the uniqueness of Bnei Yisroel. This uniqueness is based on our spiritual qualities, and such qualities have enabled us to reach heights otherwise reserved for celestial beings. We thus ask God to refrain from judging us according to natural law and strict mishpat, as this system is suited for those whose existence is earthy and relates to the material, mundane and natural order of the world. However, we ask that those who are rooted in holiness and are really not based in the here-and-now world not be judged by its strictures. Rather, the supernatural qualities of Bnei Yisroel warrant God's middah of Rachamim, as our inner potential and other-worldly qualities associate us with God in a much tighter bond, such that we are like His like His personal servants or emissaries, and we thus seek to be judged as such, reflective of our internally-holy and elevated characteristics.
The other-worldly qualities of Bnei Yisroel - as epitomized by Yitzchak and the miracles of his birth - are then contrasted in the Torah reading against Yishmael's and Hagar's performances, as critiqued by the commentators. Hagar's actions (when Yishmael was sick) were based in self-interest, while Yishmael's behavior represented the lowest, most base side of Man. The Torah then further displays the other-worldly characteristic of Bnei Yisroel when it depicts Avrohom Avinu's interaction with Avimelech. In that event, Avimelech approached Avrohom because because 'he saw that God is with him', to paraphrase the pasuk (attesting to the divine association of Bnei Yisroel). Furthermore, Avrohom used the event of his covenant with Avimelech to praise God [establishing an "eshel"], such that the mundane, earthy qualities of the story's well were associated with an elevation of the mundane to the holy. This upward, heavenly thrust to bring all to God's service represents the role of Bnei Yisroel in the world. (So, too, is it with the story of Chana. She clung to God and then dedicated her son to His service, and this association further reflects the other-worldly qualities of Bnei Yisroel which we pray will endow us with favorable judgment on Rosh Hashanah.)
However, we may never be haughty. We must always bear in mind that our unique persona and mission are divinely gifted to us, and any self-aggrandizement on our part is an utter, severe distortion of our purpose and is thus a chillul Hashem.
May it be God's will that we be continually blessed as His dearest children, and may we live up to this noble role.