It means “worry about nothing!” [i.e. “I am taking care of everything” or possibly, “leave me alone”, depending on context]

“Un rien” can also mean “something small” but not in this sentence.

ne … rien = not … anything or nothing

je ne mange rien = I do not eat anything

then it means:

Do not worry about anything!

ne … guère = not … much or not a lot

je ne mange guère = I do not eat a lot

ne … pas = not

ne … plus = not … anymore

ne … jamais = never

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

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”

or

“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”)