Although not a requirement, there is indeed an old Jewish custom to write B”H or BS”D, or to be more accurate, their Hebrew equivalents ב״ה or בס״ד, at the beginning of a letter.
What does it mean?
B”H (ב״ה) is an acronym for the Hebrew words baruch Hashem (“blessed is G‑d”) or b’ezrat Hashem (“with the help of G‑d”). Others opt for BS”D (בס״ד), which is an acronym for the Aramaic phrase b’syata d’shmaya (“with the help of heaven”).
(The quotation mark before the last letter is the Hebrew way of signalling that this is an acronym or an otherwise non-standard word.)
Reason and Origin of the Custom
This old Jewish custom has in fact led to all sorts of wild speculation about Christopher Columbus. Apparently, in almost all of his letters to his son, he included at the top left-hand corner what appears to be the Hebrew letters ב”ה, written in the Sephardic script of that era. (Yes, I was also skeptical of the claim, which even if true, does not necessarily prove anything, but there are images available online, so feel free to judge for yourself).
The idea behind this custom is that even when we go about our mundane daily tasks, G‑d should always be on our minds, our tongues and our pens. This is in line with the verse in Psalms “I have placed the L‑rd before me constantly; because [He is] at my right hand, I will not falter,” and the verse in Proverbs “Know Him in all your ways, and He will direct your paths.”
The habit of peppering our ordinary conversation with thankful mentions of G‑d was even displayed by our forefather Jacob. According to the Midrash, when he masqueraded as his unruly brother as a ruse to have his father, Isaac, bless him, Jacob almost blew his cover by repeatedly mentioning G‑d in his conversation with his blind father.
When did it transfer to the written word as well? That’s not clear. Some point to Rabbi Yehuda HaChassid (Rabbi Judah the Pious, 1150-1217) and his work Sefer Chassidim as a possible source for the custom of mentioning G‑d before writing a regular letter.
We find the use of various phrases in the letters of earlier generations. Nowadays, it has been narrowed down to B”H or BS”D (and their equivalents).
What to write? Can I throw it out?
Some are of the opinion that one should be careful of writing B”H (ב״ה), since the second letter stands for G‑d’s name. That would be a problem when discarding the letter, since G‑d’s name may not be erased or treated in a disrespectful manner. For this reason, many prefer BS”D (בס”ד).”
Most, however, rule that the letter may be discarded (although preferably not in a disgraceful manner), since the H (ה) does not stand for G‑d’s real name but for Hashem (השם), which simply means “The Name.”
The Lubavitcher Rebbe, Rabbi Menachem M. Schneerson, of righteous memory, had the custom of starting off his letters with ב״ה, as can be seen in his thousands of published letters, and he encouraged others to start off their letters acknowledging G‑d in a similar fashion.
In this way, not only are we more aware of G‑d in our daily lives, but whomever we come in contact with also gains an awareness of G‑d in his or her personal life, leading to the day when the entire world will recognize that “G‑d is one and His name is one.”