Writing ב”ה or בס”ד on a letterhead:
It is an old Jewish custom to write the Hebrew letter abbreviations of Beis Hei [ב”ה] on the letterhead of a letter, article, or document [involving even mundain matters].
The meaning of the abbreviation B”H: The abbreviation of B”H stands for “Biezrat Hashem/With the help of G-d” or “Baruch Hashem/Blessed is Hashem.”
The reason for the custom: The above abbreviations are written in order to invoke Divine assitance in one’s written activity and to show that even ones mundain activities are connected with G-d. The abbreviation is considered like the actual words themselves, and hence has the spiritual ability to arouse Divine favor.
The historical background of its inception: It is unclear as to when this custom first began. The custom is hinted to already in the writings of Rav Yehuda Hachassid and its earliest explicit source dates back to the 1500’s. Although this custom was not widespread amonst all Jewish communities, and it is not found in the letters of the Rishonim and many Achronim, nonetheless, in today’s times it has become practically universal.
The words of Rebbe Yehuda Hachassid:
The verse states “And on that day they began calling in the name of G-d” and the very next verse states “This is the Sefer.” From here we learn that when an author begins [to write a Sefer] he is to Daven to Hashem that he be successful to finish it. There was once a Sage who saw an author write the name of G-d in the top of the Sefer [Torah] and he questioned him as to why he wrote it, and the author replied as stated above. The Sage replied that one is to simply pray to G-d for assitance and not write His name as other verses speak against doing so. [This refers to a Sefer Torah, however] by other Sefarim the custom has become to write “Bisheim Hashem”which symbolizes that the person is writing Lisheim Shamayim [for the sake of Heaven] and that one who learns from it should do so Lisheim Shamayim. One is not to write on the book [the actual verse of] “And on that day..” [but rather simply Bisheim Hashem].
 Toldos Yitzchak [Uncle of Beis Yosef] Parshas Metzora p. 88 “Therefore the custom is to write in the heading of every letter B”H, as the verse states in all your ways you shall know Him”; Aruch Hashulchan Y.D. 276:28; See Shut Tzafnas Paneiach [Rogatchaver] 196-197; 303; Igros Moshe Y.D. 2:138 that so is custom; Igros Kodesh 21:5 [brought in Shulchan Menachem 5:218]; letter 1384, 7759 [brought in Menachem Meishiv Nafshi 2:650; 817]; Likkutei Sichos 6:190; 24:599; Answer to Rav Tzinner, printed in Tzadik Limelech 7:230 and Menachem Meishiv Nafshi 2:755 regarding BS”D; Heichal Menachem 1:243 and 3:57; Mishneh Halachos 6:183; Piskeiy Teshuvos 154:18
 Likkutei Sichos 6:190
 Se Mishneh Halachos 6:183
 Letter of Rebbe to Rogatchaver printed in Igros Kodesh 21:5 [and Menachem Meishiv Nafshi 818 and Shulchan Menachem 5:218]; Igros Moshe ibid
 Answer to Rav Tzinner, printed in Tzadik Limelech 7:230 and Menachem Meishiv Nafshi 2:755; Igros Moshe ibid; Teshuvos Vehanhagos 1:640; Mishneh Halachos 6:183; Piskeiy Teshuvos ibid
 The term “Shemaya/heavens” is a paronym for G-d.
 See Sefer Chassidim 884; Toldos Yitzchak ibid; Likkutei Sichos 6:190
 See Tzafnas Paneiach ibid that so is apparent from Rav Haiy Gaon, brought in Maggid Mishneh Shabbos 11:10 that even abbreviations are considered words regarding liability over writing on Shabbos; Igros Moshe ibid that even one letter of Hashem’s name contains sanctity
 See Igros Kodesh 1384 [brought in Menachem Meishiv Nafshi 2:650] that it should be written by a letterhead especially today when we need Divine assistance
 See Igros Moshe ibid that we have not found a precedent for this custom amongst our ancestors; See Igros Kodesh 21:5 [letter 7759, brought in Shulchan Menachem 5:218 and in Menachem Meishiv Nafshi 2:817] that the Rebbe asked the Rogatchaver Gaon as to the source of writing this, and in fact brought Talmudic proofs to discredit it, and that it is, and never was, customarily done in documents such as Kesubos, Gittin and loan documents. See Rav Yosef Kapach in Kesavim 1 p. 124 that this custom originally began in Europe and then spread to some, but not all, Sefaradi communities.
 See Sefer Chassidim 884, brought in Koveitz Zalman Shimon p. 71 as a source for the custom
 See Toldos Yitchak ibid who lived in the 16th century “Therefore the custom is to write in the heading of every letter B”H”; Such headings can be found on letters dating back to the 16th century.
 See Rav Yosef Kapach in Kesavim 1 p. 124 that it is clearly evident from the many letters found in archives that the above heading was not written in previous times. For example, in Yemen there are no letter which bear this heading, as the custom never spread to there. Many letters from the Rambam and his collegues have been found in archives and they also do not contain any such ehading.
 Sefer Chassidim 884
 Bereishis 4:26
 Pirush on Sefer Chassidim [Mosod Harav Cook]
 Pirush ibid