I once heard or read somewhere that Shabbos/Shabbat is one of the names of Hashem.
1- Is there a source for this?
2- If it's true, can I say "Shabbos/Shabbat" in the bathroom?
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The Chessed Lealafim (85,1) (R' Eliezer Papo) brings down Zohar that says that HaShem is called Shabbos. (He writes: דקב״ה שבת איקרי).
Chessed Lealafim says that it is not allowed to say Shabbos in the bathroom. When the Kaf Hachaim (85,7) discusses things that are assur to say in the bathroom, he says: אף לפ״ד החס״ל (even according to the da'as of Chessed Lealafim) that you can't say Shabbos in dirty places.
From the way the Kaf Hachaim quotes this opinion, it seems like theremay be those who argue, and hold that Shabbos is allowed to say in unclean places (e.g. Bath-house, bathroom, etc...). I heard in the name of Rabbi Triebitz that he said it’s permitted, but I haven't confirmed if the source is trustworthy. So please ask your Rav.
I bring the Chessed Le’alafim, quotes the Zohar on the Gemara in Berachot that says that Shabbat is one the names of Hashem. The Halacha also says that mentioning Hashem's Name in the bathroom [let alone speaking] is forbidden. Therefore, the Poskim bring down that mentioning Shabbat is forbidden.
Answer to the second part: The Yeshiva World says:
Talking Many people have a habit of talking while in the bathroom (even when one is not actually “using the toilet”). However, this practice is forbidden. Many poskim maintain that talking in the bathroom is not allowed if it is not a great tzorech when one is not using the toilet. Some people take the phone in the bathroom and talk to their friends without thinking anything is wrong with it, but this should not be done. When one is giving their child a bath and they are there for an extended period of time then talking is permitted. When one is using the toilet talking is forbidden. If one is expecting an important phone call and one could lose a lot of money by not answering the phone then some poskim are lenient. One should be careful not to talk to anyone in the bathroom since it might cause the one in the bathroom to talk back to you.
 Refer to Rashi Berochos 62a , “U’bais,” Rambam Hilchos De’os 5:6, Rama ibid, Be’er Heitiv 2, Levush 2, Mekor Chaim page 16, Machzik Beracha 3:6, Siddur Yaavetz page 87:1, Masef Lechul Hamachanus 3:8, Lekutei Maharich page 5b (old), Mishnah Berurah 3:4, Kaf Ha’chaim Y.D. 116:113, Ketzos Ha’shulchan 4:2, Minchas Yitzchok 4:61,Be’er Moshe 5:1, Doleh U’mashka page 9. The Torah L’shma 24 says it is permitted to talk in a bathroom for a tzorech and you may talk as much as you need.
 Refer to Gemorah Sanhedrin 19a, Rashi “meshum,” Darchei Moshe 3:1, Magen Avraham 1, Be’er Heitiv 1, Shulchan Aruch Harav 3:2. Although the poskim say that women talk to each to each other in the bathroom to avoid an issur of yichud there with a man, this does not apply today since we have separate bathrooms for men and woman (Bais Boruch 3:4, see Kaf Ha’chaim 3:5).
 Horav Yisroel Belsky Shlita, see Yugel Yaakov pages 1-2.
 Refer to Machzik Beracha 3:3: page 130, Siddur Yaavetz ibid, Chesed L’alafim 3:2, Mishnah Berurah 3:4, Kaf Ha’chaim 3:6, Toras Chaim page 8:2.
 Otzros Yosef 1:6, Sheiros Yosef 3:3, Yabea Omer 8:1, see Sharei Aaron 3:18.
 V’ein Lumo Michshal 5:page 31:1.
 Darchei Moshe 3:1, Rama 3:2, Levush
So you can't talk at all in the bathroom!