B"H
Often, at the top of written work, Jews write either ב"ה, which stands for Baruch Hashem =blessed is the Name i.e., God, or בס"ד, which stands for b'siyata d'shmaya =with Heavenly help.
What is the origin of this custom and why do some people choose ב"ה over בס"ד and vice versa?
I don't know the origin, but in one form or another it goes back at least to the Rambam: he began each section of his major works with the phrase בשם ה' א-ל עולם (though this has been omitted in most later printings).
There are halachic opinions that the letter ה has kedushah when it is used to represent Hashem's name (since it is one of its letters), and that therefore a paper containing such a reference must not be treated disrespectfully. (For this reason, some books, magazines, etc., use 'ד instead of 'ה when referring to Hashem.) Using בס"ד, then, avoids any such issues.
On the other hand, ב"ה has an advantage in that it is not only invoking Hashem's help (when expanded to 'בעזרת ה) but also blessing Him (when expanded to 'ברוך ה), so some people prefer that instead.
ב"ה stands for Be'ezras Hashem - which is technically the Hebrew translation for B'siyata Dishmaya.
Rav Moshe in Igros Moshe YD 2:138 writes that it is better to avoid writing bais hey and one should use beis samach dalet instead.
Rav Betzalel Stern in Batzel Hachahma 4:105 writes that one does not have to worry that a paper with bais hey will get thrown out and one can use it. He brings that the Sfas Emes and Chiddushe Harim used it on their paper(however it was divrei Torah). He brings examples from the doros harishonim that they used bais hey even on regular papers. However , he ends off saying if one uses bais samach dalet and is careful not to throw out(directly into garbage) papers with bais hey then a bracha should be upon him.
There is a discussion amongst the Rishonim whether the rule that one should greet his friend by using God's name ("והתקינו שיהא שואל שלום חבירו בשם" - Berachos 54a) is an obligation or merely a heter.
Rav J.B. Soloveitchik (also known as "the Rav") discusses1 this in הררי קדם (part two, number 124 (page 262)). The possibility that this was a heter, combined with the fact that the Rambam does not record this rule as practical halacha, convinced R' Chaim (the Rav's grandfather) that this was a temporary ruling (הוראת שעה), and that it is now forbidden to write "ב"ה" in correspondence. The Rav respectfully disagrees, based on Rashi's opinion that there is no real prohibition against using Hashem's name to greet friends.
The Rav concludes that the "new custom" of writing "בס"ד" on the top of letters is a baseless and meaningless custom ("ואין טעם ומקור למנהג זה"), based on a comparison to the laws of oaths -- an oath made in the name of Heaven, without invoking a name or reference to God, is not an oath. The purpose of this greeting is to use God's name, so referring to "שמיא" (lit. "Heaven") does not count.
1 The word "discussed" is used here in a loose sense. הררי קדם was written by Rav Michel Zalman Shurkin, based on students' notes from the Rav's lectures.
