I started tournament fishing just last year as a co-angler and had to do a lot of learning pretty quickly. Everything so far seems like excellent advice. I hope this isn't too remedial, but here's some things I discovered over the course of the year.
RODS: I usually bring 5 or 6 with me. 2 Spinning and 3 or 4 Baitcasting. That's the absolute maximum I'd recommend. Any more than 5 or 6 and you'll end up with a tangled mess, as well as difficulty getting everything stowed away during the many times your boater says "Ok, it's time to move" with about 5 seconds notice and then he's ready to leave. You don't want to make him have to wait while you're trying to get your rods put away so they don't get bounced off the back of the boat at 60mph on choppy water.
Definitely do not forget your PFD. Your boater might not always have an extra (and shouldn't be expected to). And if he does, it may not even fit properly. Also, don't forget your own fishing pliers. It's a real pain having to ask your boater to use his all the time. Also nice is your own digital scale and a set of culling tags or a culling bar. Your boater will usually have at least a scale, but there's no guarantee he'll have a second set of culling tags for you to use. And the time it takes to dig multiple fish out of a livewell to find out which one is 1.86 Lbs and which ones are 1.87 or 1.92 Lbs is time that you don't have your line back in the water.
A tackle bag that holds some 3700 size plastic boxes is likely your best bet. Maybe even swap one of them out for the same size Spinnerbait/Buzzbait box. They're awesome for keeping them organized and untangled. It's also really nice if the tackle bag has at least a couple pockets on the outisde, one for worms and another for creature baits (at least that's what works best for me). Or organize the outer pockets for whatever you use the most because they're the most accessible without having to open up the zippered part of the bag each time.
However you organize your bag and boxes, I've found it especially nice to have one box devoted to the stuff I'm going to use the most in that particular tournament. The items in it may change from one to the next, but there's not a ton of room in the back of the boat and it's nice to only have to go to one box, rather than multiple boxes and opening and zipping up the big tackle bag over and over again.
It's a good idea to ask, either before the tournament starts, or when you're on your way to your first fishing location, how your boater would like to handle when the net is needed. Some like to net all fish, while others don't want to net any at all. I've found that most will swing smaller fish into the boat, while calling for the net only when needed for bigger or lightly-hooked fish. Also, become good at netting fish! That may sound like a weird thing to say, but too many times to count I've seen someone bonk a fish right off the hook when trying to net it, rather than dipping the net low enough into the water to get underneath it. Do that to your boater when he's got a tournament-winning size fish and it'll make for a pretty uncomfortable rest of the day.
Snags: Every fisherman gets snagged now and then. Lost lures are frustrating and it can get expensive, but most boaters are pretty good about pausing a run along the bank to help get a lure unsnagged from sunken timber or inbetween rocks, etc. Do your best to keep this to a minimum though. It can get uncomfortable asking your boater for the 20th time before it's even noon to reverse the boat and go back to unsnag a plastic worm. I'd rather keep a set of clippers on me and quietly cut my own line, losing a 50-cent plastic worm and hook than frustrate my boater for yet another de-snagging run. Use your own discretion on this one. It'll be different for every body of water and what the primary baits being used are.
Good rain gear is an absolute must. I keep mine in a small cooler that has just enough room for that, plus some food/snacks (jerkey and trail mix is quick and easy) and DRINKS. Gator Ade, soda, whatever you prefer. If it's hot out, your boater will often have one of the storage bins filled with ice. They're usually fine with stowing your drinks in there if you ask.
I can't stress this one enough. Keep a set of clean, dry clothes ready and waiting in the car when you get back. You'll thank yourself when you've been fishing for 8 hours in the hot and humid middle of summer, you're drenched in sweat and you have no dry clothes to put on for the 3 hour drive home. Or even worse, it's cold and rainy and you're soaked and have to drive home in that. There's also the thankfully-rare but real possibility that you could fall out or get bounced out of your boat and into the water if the weather is particularly rough. If you have dry clothes to change into back at the ramp, it could mean the difference between a 15 minute delay for you to change clothes, vs. your tournament being completely over and all that time and money wasted.
After the weigh-in is usually the best time to discuss gas $s with your boater. You'll only then know how far you've had to run over the course of the day. It may be 20 miles or you may end up in view of the boat launch the entire time. If you're unsure, simply ask "What are you thinking for gas?" and he'll usually come up with a number. I like to add an extra five bucks to it, whatever it is (unless on the VERY rare occasion you might get some jerk who's trying to screw you out of money). Luckily, those types don't seem to last long or go very far, as they quickly get a bad reputation. Some will even refuse to take any money at all. Either way, you'll quickly get a good feel for how much or how little gas your boater will end up using.
Last but not least, after the tournament is over, help your boater clean up his boat. Don't leave any messes, pieces of soft plastics, worms, fishing line, or hooks stuck in his carpet. If he's wiping down the motor and outside of his boat, ask him if he has an extra towel and if he'd like you to help him out. It only takes a few minutes and you'll be known as one of the good co-anglers to draw.
I've probably forgotten some things, but that's what comes to mind right off the top of my head.