The Souls franchise, birthed by the brilliant minds of FromSoftware and Hidetaka Miyazaki, has provided an ample amount of influence and difficulty unmatched in the industry. Despite the franchise's immense acclaim, a dispute over the title's underlying meaning still exists to this day. From Demon's Souls to Dark Souls, then on to its later sequels, the franchise has been prone to evolution. In addition to the other various franchises under the FromSoft banner, fan conjecture can often become convoluted, especially when attempting to discern what exactly it means to be a Souls game.
To totally understand the namesake, it's important to look back on its initial creation. It all started in 2009, when From Software's Demon's Souls became a bit of a cult hit. Blending horror tropes and Berserk influences with tight action-RPG gameplay, Demon's Souls turned heads almost immediately. But, what was next? With the Demon's Souls IP tied up with Sony, FromSoft had to alter the name in order to bridge the franchise to all platforms and audiences.
Enter Dark Souls, born through a few bouts of complicated retitling. At first, series creator Miyazaki wanted to call the sequel to Demon's Souls "Dark Race," which most certainly would have raised a multitude of red flags. FromSoft's next two potential titles were "Dark Lord" and "Dark Ring," the latter of which the team eventually went with. Little did From know, "Dark Ring" is literally a slang term in England for "anus," so it was high time the developer got creative. Instead, however, they kept the Souls mantra and rid it of the Demon in favor of a more ambiguous interpretation.
That said, there is actually an in-lore reason behind the name. As per the Dark Souls Wiki:
The Dark Soul, one of the unique Lord Souls, was claimed by the Furtive Pygmy, the ancestor of the Chosen Undead and Humanity. In the introduction to the first Dark Souls, it is depicted as a small, reddish flame like a candle. [...] During the advent of Fire, the concept of Disparity came into existence, and with it the Dark and the Light. From the Dark came beings (Hollows) who claimed souls near the First Flame; 4 of these became the Lords who claimed the mightiest and most brilliant of souls.[7] While the other three "Lords" chose larger souls, there was a fourth, a lowly and forgotten being known only the Furtive Pygmy. His soul was the Dark Soul.
Basically, the Dark Soul is the narrative impetus behind the entire Dark Souls franchise - which checks out, considering it's in the title and everything.
And if this isn't enough, there's an entire Steam Community guide dedicated to the intricacies, metaphors, and astute observations surrounding the franchise in its fullest form.