What did Le'ah mean when she proclaimed "Ba Gad"?
Rashi #1, Rashbam and Targum Yonasan: She meant that she merited good fortune 1 (Mazal Tov) with the birth of Zilpah's son.
Rashi #2: She was referring to the fact that Gad was born circumcised. 2
Seforno: She was saying that this baby came by chance. She only offered Zilpah to Yakov because she happened to stop bearing children. 3
Tosfos ha'Shalem (6): The good fortune that I had to have sons, and it stopped, returned to me.
Eliyahu comes from Gad (Hadar Zekenim, from Bereishis Rabah). Da'as Zekenim says that he was a Kohen, but his mother was from Gad.
Why is "Ba Gad" written as one word?
Rashi: She said that Yakov 'Bagad' (betrayed her), through accepting her offer and marrying her maidservant. 1
Moshav Zekenim citing R. Shmuel ha'Chasid: Shevet Gad 'Bagad' Eretz Yisrael, and opted to live in Ever ha'Yarden. 2
Moshav Zekenim: Bagad is an expression of telling [that Leah will become pregnant again].
Tosfos ha'Shalem (1, citing R. Efrayim): It is written lacking an Aleph 3 , to teach that Bnei Gad would (be strong warriors, but) not be great in Chachmah.
Asarah Ma'amaros (Chikur Din 3:18): Leah betrayed Yakov (refer to 30:9:2:3).
Tosfos ha'Shalem (4, citing R. Efrayim): He was covered with skin (a fetal sac), like a Beged (garment). This is a good sign for children.
Tosfos ha'Shalem (4, citing Riva Agiloni): Bagad is an expression of telling. Eliyahu comes from Gad, and he tells the future and hidden matters.
Hadar Zekenim rejects this, for Leah asked him to do so. Tosfos ha'Shalem (9) explains that Yakov should have refused to lie with her Shifchah in place of her, for Leah has many children. She did not call her son Boged, to avoid publicizing this.
Moshav Zekenim asks why there is no such Drashah about Reuven and [half of] Menasheh, who also opted to live there! Tosfos ha'Shalem (10) says that Gad first asked to inherit there. (Reuven merely followed Gad. We always say "Bnei Gad and Bnei Reuven", even though Reuven was more esteemed and the leader of its Degel - PF.)