Does this sentence:
Ne vous préoccupez de rien !
Do not worry about anything! ≠ Worry about nothing!
Do not worry about the smallest things! (Those things are nothing important.)
I’m not sure how to interpret the rien bit.
To understand better what rien means, knowing its etymology helps.
Rien comes from the Latin res that means "thing, affair, stuff": Res publica → public matters.
The fact it is essentially used in negative sentences shifted its perceived meaning (nothing) to be exactly the opposite of its original one (something).
Ne vous préoccupez de rien ! literally translates to "Don't worry about anything!"
The following suggestions are not literal translations at all (and only address the meaning of "rien" indirectly by using its opposite, "everything"), but if I saw “Ne vous préoccupez de rien” used as a slogan for a hotel, travel agency, or even a DUI law firm, I would take it to mean:
“Relax, we’ll take care of everything”
“Relax, just leave everything to us”
Regarding your second interpretation (essentially: “Don’t sweat the small stuff”) it occurs to me that it is not really that illogical, but it would require a “pas” to make the “rien” mean what you propose in Option #2 (otherwise, it’s just part of the “ne/rien” construction).
It would also require some extra words and letters (including the pluralization of “rien”) to make it an understandable, if not idiomatic, way of saying “Don’t sweat the small stuff”:
“Ne vous préoccupez pas des petits riens.”
It would still, however, not be as idiomatic as:
“Ne vous préoccupez pas des détails” (with "rien" not present at all) or better still
“[Ne] t'inquiètes pas pour les détails/ [Ne te] casse pas la tête avec les détails." (which wouldn't use either "préoccuper" or "rien")
In that particular context, it means not to worry about a single thing (nothing). Even though it could translate into
anything, anything would imply nothing as well because not worrying about anything , is not worrying at all.
anything = a thing of any kind
Do not worry about
anything = do not worry about a thing of any kind = don't worry at all