I personally don’t require a boom op to bring anything but their own headphones. I supply the rest. That said, everyone has a different way of doing things, and I will never stand in the way of someone using their own techniques and supplies if that’s what they are comfortable using. I provide a variety of expendables and lav mounts, lav bullet, boom poles, tapes and stickies and etc. But there are all kinds of things available these days, so I’d say that if you were looking to put together a little kit, get yourself some moleskin, molefoam, some kind of furry overcover set, transpore, topstick, maybe some Joe’s sticky stuff, preferred lav mounts, a headlamp, lav bullet, and probably a boom box or chair with the ability to mount the boom to. K-Tek makes a little fanny pack that looks like a decent pouch to keep things in. I ordered some stuff from a Sound Hard and they sent me a complimentary fanny pack that is sort of the same thing, so i keep that stocked for my boom ops.
But if you think you’ll need gloves, a bandana or sweat rag, multi tool, or any other such thing, it’s better to be prepared and not need it than to be empty handed when you do need something.
I recently worked in a dusty warehouse where there was years of pigeon poo all over the place, so I bought anti bacterial medical masks for the whole crew. Now I have those in my kit too.