I was amazed when suddenly Yacov Avinu asked his brother for the right of being first born son when Eisav asked for food to eat. Why did he choose that moment?
The following possibility occurred to me, but I haven't seen it written in any sefer. The Midrash Rabbah in Bamidbar 4:8 states that the right to perform the avodah originated with Adam and was transmitted through subsequent generations from individual to individual (either to a firstborn or to a tzaddik, depending on the circumstances). It was also originally linked with special garments that were given to Adam, which were also passed along.
There are sources (Pirkei Dr"E chap.24 and others) that Nimrod somehow ended up with these garments, and that Eisav killed him and took them for himself. I believe there is also a source (don't recall where) that when Eisav met Yacov that day he was exhausted from having just killed Nimrod. The aforementioned Midrash also mentions that Yacov wanted the birthright from Eisav in order to obtain the Avodah rights which generally went along with it. Rashi (25:31) writes that Yacov thought to himself "This wicked person is not worthy of bringing offerings to God".
If we put all of these pieces together, there emerges the following possibility: Yacov saw Eisav carrying (or perhaps even wearing!) the garments of Adam used for the Avodah after having just gotten them from Nimrod. This brought into stark focus the fact that Eisav would likely receive the rights to bring offerings to God (as he was the firstborn), and this thought repelled Yacov and inspired him to immediately go about stopping that from happening by acquiring the firstborn rights from Eisav.
R. Yosef Bechor Shor's explanation of this incident accounts perfectly for the timing oof Yaakov's request. He explains the incident as follows:
Eisav was a hunter, and as such would often go hunting in the forest for several days at a time, becoming prone to hunger and thirst. When the Torah describes Eisav as עיף (tired), it doesn't simply mean that he was a little tired. This is a description of the perilous situation Eisav was in at the moment. After going on a hunt for several days Eisav was on the brink of death. He was in fact so exhausted that he was not even capable of lifting his hand to his mouth to feed himself. He was far from the city, and any delay in food getting to him would kill him. At that moment Yaakov was out shepherding the sheep when he came upon Eisav. Eisav asked Yaakov to pour him the food (הלעיטיני נא) because as mentioned he did not even have the strength to feed himself.
At this point, Yaakov made the following calculation: if Eisav would die here on his own Yaakov would be the sole inheritor of everything. If he would feed Eisav thereby saving him from death Eisav would remain as the firstborn and get everything. So Yaakov told Eisav that he doesn't want to save him if it will cost him such a loss. But if Eisav would sell him the birthright then Yaakov would keep everything and suffer no loss by saving Eisav. Thus, Yaakov gave Eisav a choice. Either give him the birthright and be saved, or don't give him the birthright and die of hunger/exhaustion. If Eisav would not care enough about his own life to sacrifice the inheritance, Yaakov would certainly not have to care enough about Eisav's life to sacrifice the inheritance.
Thus, the connection between Eisav asking for food and Yaakov asking for the birthright becomes abundantly clear. It was only because Eisav needed Yaakov to save his life that Yaakov was able to force Eisav to give him the birthright. Yaakov never brought up the birthright at another time, because there never was another time when he had something to hold over Eisav as he did in this case.
והוא עייף. דרך הציידים להיות עייפים ורודפים אחר החיות ופעמים טועים ביער שלשה וד' ימים [ויגיע] להם רעב וצמא וכן היה לעשו עד שהגיע לשערי מות ולא היה יכול להשיג ידו אל פיו כמו שהוא אומר
הלעיטיני נא וגו'. כלומר שפוך לתוך פי ואני אוכל שאיני יכול אפילו לתת לתוך פי ולא ללכת אל בית אבי אך מצא יעקב אחרי הצאן במרעה רחוק מן העיר אם היה האוכל מתמהמה לו הי מיד מת ולכך אמר לו יעקב אם אתה מת מיתת עצמך הרי כל אשר לאבי בידי וכל בני קטורה וישמעאל ולוט שמילט זקיני מן השבי ומן המות יהו משועבדים לי ואם אוכילך ותחיה הריני מפסיד שרות גדול וממשלה גדולה שאתה בכור ותקח הכל אבל אם תמכור לי בכורתך אז אוכילך ותחיה ולא תמות ואני לא אפסיד במה שאוכילך ואם לא תחמול על עצמך בשביל שלא תרצה שלא תבא לי הממשלה גם אני לא אחמול על עצמך כי כמו שאתה רוצה לעצמך אני אוהב לעצמי ואיני אוהבך יותר ממני ומחמת טענה אמר לו יעקב
מכרה כיום את בכורתך לי. כלומר שתעמוד במכירתך לעולם כמו היום