I am coming to this very late. I found it amusing that there are so many comments about what exactly is an inspiration and it's use as a tag. This is quite typical of comment forums where the original topic is quickly diverted to a discussion of something somebody mentioned that has little or nothing to do with the actual topic. Of course, now I'm doing that myself.
I think Herbert was most inspired by his imagining of the Bene Gesserit, who they were, what they were about, what was their agenda and motives, and how did they do what they did.
Herbert was masterful at character development and in Dune this extended to developing the major competing organizations of power whose interactions provided the backbone of the plot which he so capably fleshed out with numerous supporting characters, also well developed with their own significant subplots.
It was extremely important to him to make the world of the Bene Gesserit at complete as possible so that their interactions with the other main power organizations, dealings both large and subtle, seemingly important or seemingly minor, were seen by the reader as the multi-leveled, plots within plots that made the book such a classic.
I believe Herbert created the litany against fear to both reveal and substantiate things he wanted the reader to know about the Bene Gesserit. Although it could easily be posited and has been in this forum, that there are elements of Zen and perhaps Islam in the text itself I sense that those influences were secondary to his desire to his goal of revealing the nature of the organization.
The litany's main purpose is to tell us about a primary goal of the inner organization. The Bene Gesserit are a sort of militant quasi-religious order on steroids. They have been following a specific agenda for millennia. They develop young women into virtual manipulation machines, skilled in any number of ways to influence men and families of power at the highest level. This way they can remain relatively secret and anonymous while wielding tremendous political power. One important educational goal of the order was to isolate and develop the human and disassociate the animal nature of its sisters. It sought in both it's breeding program and its training to create beings who were completely human with no animal influence. This had multi[le purposes from training the sisters to perform extraordinary feats of mental and physical control and acuity and for creating impenetrable bonding and discipline within the whole of the organization.
The Litany is a tool the Bene Gesserit uses to focus it's sisters, through its vocalized repetition, back on their humanity when faced with fear, which they knew to be animal in nature and therefore highly inferior and unwanted. Thus the "little" death. They also knew that although fear is little and animal it is still powerful enough to kill the mind of a human and lead to total obliteration.
By facing our fear we separate the human from the animal. We let it pass over and through us becoming transparent and invisible to it, robbing it of its power through sheer human mental superiority. When it passes through we do not go merrily back to whatever we were doing but we use the experience as a learning moment. We turn our mind's eye back inward to see fear's path. We see it's intentions, where it planned to hurt us, what vulnerability, what human weakness it was to exploit and how it planned to do so.
But there is no substance there. We remember that fear has no substance and only we remain, more human than ever.
The Litany tells us a lot about the Bene Gesserit in a short time. This was Herbert's intention and he was inspired by his own imagination.