What things should a skater carry when they skate? Like all questions phrased like this, the answer is "it depends on what you're doing." Here are some items that I suggest carrying around, why it's a good reason to carry, and what situations they are particularly good for.
You don't need to own big hardware like a full bearing press. Your local rink will have one that you can use.
A nut driver/skate tool is probably the handiest thing to carry. They are used for swapping wheels and adjusting action on traditionally designed plates. Standard American size is 1/2 inch and the standard international size is
Alen keys can be used to (1) adjust the pinch on toestops and (2) disassemble Arius Platinum plate.
Sweat on the skate floor can be a slipping hazard and contributes to those dirty patches on skates (when skating rinks are humid (especially when there are fog machines)). You can wipe down intermittently or use it as a headband.
Band-aids
Antibiotic ointment (Neomycin/polymyxin B/bacitracin (Neosporin))
Ibuprofen
You never know when you or someone around will need a little support.
Even if you never use them, it's good insurance
Spare nuts (x8) - If you're crazy like me and change wheels a lot, you're bound to loose one
Spare bearings (x16)
Spare spacers
Spare speed rings - some wheels need a little clearance
Now this really depends on if you skate a lot. I've needed to replace y laces once before; they broke because I had laced up so many times! Perhaps salt from sweat accelerated the lace's degradation. All that to say, a spare set of laces might be a good idea to pack when your laces have some age to them.