What did Leah 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.
Bereishis Rabah (91:9): Eliyahu will come from him; he will Gaded Mashtisan (cut off) of idolaters. He was "mi'Toshvei Gil'ad" (Melachim 1:17:1), and "all Arei ha'Gil'ad" belonged to Gad (R. Yehoshua 13:25). 4
Malbim: My Mazal returned a second time. I look forward to giving birth again.
Why is "Ba Gad" written as one word?
Rashi: She said that Yakov 'Bagad' (betrayed her), through having Bi'ah with her maidservant.
Moshav Zekenim citing R. Shmuel ha'Chasid: Shevet Gad 'Bagad' Eretz Yisrael, and opted to live in Ever ha'Yarden. 1
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 2 , 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 3 , and he tells the future and hidden matters.
Ha'Emek Davar: It is an expression of Nisu'in, 4 like "b'Vigdo Bah" (Shemos 21:8); we expound, he put his garment over (married) her (Kidushin 18a-b). As two words it is the Brachah of Mazal that people give at Nisu'in.
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.)
Ha'Emek Davar: Leah was moved by the Nisu'in, which she arranged for Zilpah's benefit, therefore she named the baby based on the Nisu'in.
Why does it say "Es Shemo' only for Gad and Asher?
Tosfos ha'Shalem (11:10): Their children were anointed with oil more than other Shevatim. 1
Pesikta Zutrasa on this verse says that daughters of Asher were beautiful, and married Kohanim and kings, who are anointed. I did not find a source for Gad (PF).
QUESTIONS ON RASHI
Rashi writes that Yakov betrayed Leah through having Bi'ah with her Shifchah. Leah gave her to him!
Tosfos ha'Shalem (9): 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.