Oops, sorry, it's about an inch and a half across, or roughly the size of a large walnut.

A box in which to store his STONE

That's a vinaigrette, I believe. Small pieces of natural sponge were soaked in perfume, and kept therein. It was used to mask the whiff of the city streets.

snuff box or salt/pepper shaker.

Ah of course, what adeteal said...containing anti-plague whiffy essences in a sponge??

Probably utterly nuts, but it looks a mini cheese grater - for his beloved parmesan?

I have a less elegant box like this that I keep my sole gold sovereign in.

@janetravers and @adeteal are the closest so far.

Got it! I think. A nutmeg storage/grater thingy? Wasn't nutmeg really valuable?

Exactly, @janetravers. must be for nutmeg, no?

Pomander?

Yay, we have a winner: @janetravers with nutmeg grater. This one's from about 1690, and the tulip is just a topical decorative theme. As any fule kno, the only way to have nutmeg is freshly grated, so you just whipped this little box out at the table and seasoned your food with your very valuable whole nutmeg. Maybe you offered it around the table, and maybe you didn't. That was quick. The next one's going to be hard.

Woohoo! Do I win a nutmeg grater, then? ;) No need, my great victory is enough for me. :)

That was GRATE!!! *scarpers sharpish*

Some form of pomander to detract from the smells