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 Yaakov 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 (71:9): Eliyahu will come from him; he will Gaded Mashtisan (cut off) of idolaters. He was "mi'Toshavei Gil'ad" (Melachim I 17:1), and "all Arei ha'Gil'ad" belonged to Gad (see Yehoshua 13:25). 4
Malbim: My Mazal returned a second time. I look forward to giving birth again.
Rashi: Like the Pasuk in Daniel, "Godu Ilana - cut down the tree!" (Daniel 4:11). Bartenura asked the source, and brought this verse. This implies that it was not in his text of Rashi. Perhaps his answer was included in later editions of Rashi. - PF)
Seforno: And it is written as one word, because it is rooted in the word 'Bagad,' not with reference to Yaakov (as Rashi explained - Refer to 30:11:151:1), but because her ability to bear children had ceased (like "Bagdu Chemo Nachal" - Iyov 6:15).
Bereishis Rabah: Chachamim were arguing whether Eliyahu comes from Gad or from Binyamin. He appeared to them, and said that he is from Rachel. Tosfos (to Bava Metzia 114b DH Mahu, citing Eliyahu Rabah 18) - he proved this, for it says in the lineage of Binyamin "v'Ya'areshyah v'Eliyah" (Divrei ha'Yamim I 8:27) Pane'ach Raza (Vayigash) - this is unlike the opinion that Eliyahu is Pinchas. Or, his mother was from Binyamin [or Gad]. (The Torah tells us that Pinchas' mother was from Putiel (Shemos 6:25), i.e. Yisro and/or Yosef (Rashi loc. cit). (CS)) Aderes Eliyahu (of the Ben Ish Chai, Parshas Pinchas, citing the Ari Zal) - he is attributed to different Shevatim due to sparks of his Neshamah that came from them.
Why is "Ba Gad" written as one word?
Rashi: She said that Yaakov 'Bagad' (betrayed her), through having Bi'ah with her maidservant.
Moshav Zekenim citing R. Shmuel ha'Chasid: Shevet Gad 'Bagad' (betrayed) Eretz Yisrael, and opted to live in Ever ha'Yarden (see Bamidbar 32). 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 Chochmah.
Asarah Ma'amaros (Chikur Din 3:18): Leah betrayed Yaakov (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'amek 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 Berachah of Mazal that people give at Nisu'in.
Moshav Zekenim asks why there is no such Derashah 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'amek 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 Yaakov betrayed Leah through having Bi'ah with her Shifchah. But Leah gave her to him!
Tosfos ha'Shalem (9): Yaakov 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.