To my knowledge, there is no halachic source for this specific practice. It's just something that people do to acknowledge Hashem's role in their endeavors (whatever they understand that to mean). Generally speaking, this seems to be a good practice. Philosophically (though probably not halachically), it is fulfillment of the what the Rambam writes in Hilchos Berachos 10:26:
The sum of the matter: a person should always cry out in prayer about the future and ask for mercy, and express gratitude for the past, and give thanks and praise according to his ability. The more a person thanks and praises Hashem continually, the more praiseworthy.
Over the past few years I've become an increasingly big fan of the Ralbag's commentary on Chumash. I began noticing the Ralbag's use of a mysterious acronym: בג"ה. He seemed to use it in the same way that other people use בס"ד, but I couldn't figure out what the letters stood for. Finally, while using an edition of the Ralbag with expanded acronyms, I realized that בג"ה stands for בגזירת הצור (b'gezeiras ha'Tzur) meaning "with the decree of the Rock."
I'll admit right up front that I don't know why the Ralbag prefers בג"ה over בס"ד. I'd venture to guess that his preference for this phrase can be traced back to his metaphysical premises, and I wouldn't be surprised if this decision reflects some controversial view about the relationship between God and the universe of which I am unaware. Frankly, none of that really interests me for the purpose of this post.
The reason I am writing this post is to explain why I prefer בג"ה over בס"ד. It is possible that my reasons apply only to myself, but I want to share them in order to see what others think.
My first reason has to do with the use of the term "gezeirah" (decree) as opposed to "siyata" (assistance). The metaphor of gezeirah brings to mind the idea of Malchus Hashem (God's Kingship). It also highlights the lawful, objective, and just manner of Hashem's governance of His world: "ki kol derachav mishpat" - "for all of His ways are justice" (Devarim 32:4). The term "siyata," on the other hand, caters to a more egocentric way of viewing Hashem's role in our lives. Instead of conceptualizing Hashem as the Melech Whose decree governs the entire universe, of which I am but a small part, the term "siyata" frames Hashem as "The One Who Helps Me." In other words, "gezeirah" emphasizes our acceptance of Hashem's Malchus, whereas "siyata" emphasizes His servicing our needs
(To reiterate: this post expresses my views about why I prefer בג"ה to בס"ד. I'm not claiming that the associations I have to these terms reflect their only meaning, nor am I suggesting that everyone else thinks the same way I do. I'm just telling you what I think about when I encounter these terms. The ideas to which I am referring are objective truths, but the aim of this post is to express my subjective relation to those objective truths.)
My second reason for preferring בג"ה over בס"ד is that to my mind, siyata implies hashgachah pratis specifically, whereas gezeirah includes both hashgachah pratis (individual providence) and hashgachah klalis (general providence - i.e. natural law). In other words, the notions of "help" and "assistance" imply intervention in a natural order. From this perspective, בס"ד focuses on Hashem's role as the One Who intervenes in the natural order. "Gezeirah," on the other hand, refers equally to general decrees of hashgachah klalis - the laws of nature, which Hashem created and maintains - as well as the particular decrees of hashgachah pratis. As such, בג"ה is more all encompassing than בס"ד.
My final reason for preferring בג"ה over בס"ד is the use of Tzur (Rock) in the former. The Rambam explains the meaning of this mashal in the Moreh ha'Nevuchim 1:16. For the sake of brevity, I have omitted many of the prooftexts cited by the Rambam:
Tzur is an equivocal term. It is a term denoting a mountain. It is also a term denoting a hard stone like flint. It is further a term denoting the quarry from which quarry-stones are hewn. Subsequently, in derivation from the last meaning, the term was used figuratively to designate the root and principle of every thing ... On account of the last meaning, God, may He be exalted, is designated as the Rock, as He is the principle and the efficient cause of all things other than Himself ... The verse, "you may stand firmly on the Rock" (Shemos 33:21) means: rely upon, and be firm in considering, God, may He be exalted, as the first principle. This is the entryway through which you shall come to Him, as we have made clear when speaking of His saying [to Moshe], "Behold, there is a place by Me; [you may stand firmly on the Rock]" (ibid.)Thus, when we frame Hashem as Tzur in בג"ה, we remind ourselves that everything that happens is a decree emanating from the Cause of all causes - the Foundation of all existence. In contrast, the phrase בס"ד stops short at shamayim - a reference to Hashem's creation, and not Hashem, Himself. Sure, we imply that we are really referring to the Creator of shamayim - but why not go the full distance and make explicit reference to Hashem as such? (Incidentally, this is one advantage of using ב"ה over בס"ד).
For these three reasons, I prefer the Ralbag's בג"ה over the conventional בס"ד. While both phrases accomplish their intended goal, בג"ה brings me closer to ahavas Hashem (love of God) and yiras Hashem (fear of God) than בס"ד.
The only catch is that nobody but the Ralbag uses בג"ה! I guess we'll have to see what we can do about changing that.