The Declaration ( Viduy )said when bringing the BIKKURIM includes saying “An Armenian wanted to destroy my father, and he went down to Egypt” Yaakov had many things Happen to him why choose the episodes with Lavan and going down to Egypt (Mitzrayim) versus his battles with his brother Eisav?
I have to check my source, but I believe a simple answer was that Lavon had desired to wipe out Yakaov and his family. This would have terminated the nation of Israel. This is the first instance of Hashem's divine intervention (Hashgacha Pratit) to save the Jewish nation.
As a side note, we tend to get excited about the sensationalism of Yetzias Mitzraim, the exodus from Egypt. The revealed miracles of the plagues and splitting of the sea. We also need to realize the hidden miracles of Hashem's divine intervention in the laws of nature to maintain the nation of Israel throughout history.
Thanks for considering this approach.
The Alshich answers that Lavan is the source of all our problems. If he would have given Rochel to Yaakov then Yosef would have been born first, and the brothers would never have fought with him. Thereby never selling him to slavery, this would have prevented our exile in Egypt from ever happening. That is why he is mentioned as he was the source of our going down to Egypt and not Eisav.
Reb Yaakov Kaminetsky says it is not just Lavan it is all Armenians. As when Yaakov was going to marry Leah and not Rochel, they all knew and said nothing.
Rav Kasriel Auerbach of Ponovezh explains there is a fundamental difference between Eisav's hatred of the Jews and Lavan’s hatred. That is Eisav hatred in this time period, was based on the fact that Yaakov had stolen his Brachos, as later we see when they meet up again and Yaakov points out that Brachos that Yitzchak had given had not come true Eisav does nothing, but Lavan should have had nothing but positive feelings for Yaakov as he made him rich and was the reason he had sons. This is why we focus in this declaration on Lavan so we understand and internalize that the eternal Hatred of the Jew is one without reasoning.
The Brisker Rav has a different reason. The declaration of Bikkurim is meant to thank Hashem for a deliverance from hardship. Eisav is still around and still can cause us problems while Lavan is no longer around so in our declaration we are thanking Hashem for delivering us completely from Lavan but Eisav as an enemy to the Jews still stands.