Wikepedia talks mostly about its use in medieval Europe, France especially, and says that it originated with the Franks along the river Leie, but also mentions offhand that it appears in Indian and Scythian art centuries before the Franks were an entity at all. Are those symbols connected or is it just a coincidence that they look the same? And if they do have common ancestry, what is it?
Just because I happen to have it open in another tab, here's [Parker's Glossary of Heraldry on the fleur-de-lis](http://www.heraldsnet.org/saitou/parker/Jpglossf.htm#Fleur de lis):
Although there has been much controversy concerning the origin of this bearing, no doubt it represents the lily, but in a conventional form, such as was produced by the workers in metal. It is essentially the Royal Badge of France, having been adopted by King Louis VII. in the twelfth century, in allusion to the name lois, or lys. It appears amongst the Royal Badges in England in the time of the STUARTS. From some of the following examples it will be seen how variously the name is written in ancient rolls of arms. It will also be observed that the fleur-de-lys is subject to certain variations, e.g. stalked, slipped, leaved, seeded, and even fitchy.
So... as far as one expert on heraldic symbols goes, it's a mysterious symbol that probably came from metalworking, or wrought-iron representations of "the lily."
And it's used in some highly variable ways (the entry goes on from that quoted bit).
Given that heraldry is full of things like mullets - which look like stars, but are really a stylized shape descended from "the rowel of a spur," the thing you'd strap on to your heel and dig into a horse's side to make it gallop faster - I'd be entirely unsurprised if it simply descended from some kind of common tool or weapon-part that gradually became more stylized and lily-like.
My vote would be for a spearhead or a poky bit on the end of a fencepost - easy to make, functional, and kind of shapely. "Fitchy" in the above description means "with the downward end pointy" or even "forming the head of a spike," if you like. And the entry goes on to describe elements in some arms being "fleury" - terminating in a fleur-de-lis.
But I've got no sources on that. It's sheer speculation.
It's also more commonly depicted as a lily (a literal flower) or a lily shape.
You might get some joy trying to trace a similar coincidental(?) relationship between the fylfot, a device found in medieval church windows and I believe in pre-Christian Celtic carvings, and the swastika, as it appears in Buddhist and Hindu temples.
I never could - it seems to be just a natural geometric pattern that craftsmen naturally "discover."