Why did Lavan add "Elohei Avihem," and why did Yaakov swear by Pachad Yitzchak?
Seforno: He was explaining that he appointed the god of Nachor together with the G-d of Avraham, because he was also the god of Terach, who was the father of them both. However, Yaakov swore by the G-d whom his father Yitzchak feared, 1 for Yitzchak was not the son of Terach.
Tosfos ha'Shalem (4): Yaakov swore in the life of his father Yitzchak, who did not normally swear, so this was a great matter. 2
Ramban: Pachad Yitzchak is Hashem's Midas ha'Din. Ha'amek Davar - He will punish Lavan severely if he transgresses.
Malbim: Lavan summoned Elokei Avraham, who knows hidden matters, to judge about them (Inuy), and Elohei Nachor, whom he thought oversees revealed matters, to judge open matters (war). He added Elohei Avihem, to justify Elohei Nachor, for he was also the god of Terach, the father of Avraham and Nachor. Yitzchak refused to join another matter with Hashem, so he swore by Pachad Yitzchak, i.e. the Mitzvah of Kivud Av and his fear of his father.
Ha'amek Davar: The Bris against war applied also to Lavan. He mentioned also Elohei Nachor, to show that he was not concerned for Elokei Avraham, since he does not serve him. This pained Yaakov, but to swear only in the name of Elokei Avraham would impinge on Lavan's honor. Therefore, he swore by Pachad Yitzchak (refer to 31:53:151:2.
Oznayim la'Torah: Lavan referred to 'Elokei Avraham' - whose Midah was Chesed 3 and who entered into a (positive) covenant with Avimelech and swore to him that they would do good to each other. 4 But that was not the gist of the covenant here, which entailed not doing harm to one another, 5 which was similar to the covenant that Yitzchak entered into with Avimelech. 6 Which explains why he swore by 'the [G-d] Pachad of Yitzchak his father.'
Why did Yaakov swear and not Lavan?
How could Yaakov accept an oath in the name of idolatry? "Lo Yishama Al Picha" (Shemos 23:13) forbids this!
Moshav Zekenim: Perhaps Yaakov knew that [Nachor] did not serve idolatry (so Elokei Nachor is Hashem). 1
Yaakov did not ask Lavan to swear. Lavan mentioned Elokei Avraham and Elohei Nachor on his own. Yaakov deviated, and swore by Pachad Yitzchak, to show his disapproval (based on Malbim, Ha'amek Davar).
If so, when a scribe writes Elokei Nachor, he must intend that it is Kodesh! Lavan said also Elokei Avihem. Terach was an idolater! However, he repented (refer to 15:15:2:1); perhaps 'Elokei Avihem' is Kodesh, based on his final belief. Maseches Sofrim (4:7) and Maseches Sefer Torah (4:4) say that both of them are Chol. Bereishis Rabah (74:16) and other Midrashim say that Elohei Nachor is Chol, and Elokei Avihem is Kodesh (Matanos Kehunah - due to Terach; Etz Yosef - for it includes Avraham), and Chol (PF).