Super important topic. One you'll likely see come up often now that it's summer time.
Let's assume that hydration and sun protection (which is what he meant by hat, right? If not, please wear sunscreen) is a given, as the OP suggested. Other than that, there appears to be controversial information floating around here. Disclaimer: I'm not a medical expert, just some guy who lives in Florida who likes to try and make sure his information is correct.
1.) Bring a cooler filled with ice and some water (I like the medium-sized soft coolers with a shoulder strap). Fix the cooler to the fence using the shoulderstrap and a carabiner (or whatever you have. Bungies work too). The goal here is to keep the cooler off of the ground and other flat surfaces (like uncovered benches). Elevating the cooler will dramatically increase the length of time the water stays cool. I see so many people show up without this. This is how you keep your beverages and your towel cool. Don't skip this step. Oh, hopefully it's obvious, but put all the beverages you can fit into here. Rotate if you need to, as room is made.
2.) Cold towels around your neck (and/or wrists) will have a cooling effect on your body's temperature (1). Not only will it cool the blood running through major arteries, it also produces a psychological effect of feeling cool and fresh. Just remember to towel off after the wet and cold towel.
3.) Sweating is your body's natural cooling mechanism, but it's poorly understood in general. For the sake of brevity, heat is transferred from your body, via the skin, to the layer of sweat. The layer of sweat absorbs heat until it's hot enough to evaporate. This process effectively transfers the heat from your body to the environment
(2). Some will use this to suggest that toweling off the sweat is counter-productive, but I always push back.
4.) Towel off the sweat during change-overs (after the cold and wet towel, of course). Once the heat has been transferred to the sweat, get rid of it! Your body can't count on you having a towel, or a nice breeze to help get that water off you,
so it relies on evaporation. So, take a cue from nature, and evaporate that heated sweat with a towel! This lets the process start over with a thin layer of sweat and some dry area for sweat to quickly cover again.
5.) Change wet shirts in between sets. Once a shirt becomes saturated, it sticks to your skin smothering the pores, reducing their ability to sweat. Shirts that wick sweat are essentially creating a path for the sweat to evaporate through, but even they can become super-saturated. You have multiple shirts, so just bring a few and change. Easy.
6.) Wear high-performance underwear! This sounds silly, but it makes a huge difference (in my experience). It helps you stay dry-ish down there, which in turn, helps that part of your body stay cool. In this category, I'm just going to assume that you're all wearing a complete tennis outfit, wear
all of the clothing is designed for extremely hot conditions.
7.) Take your change-over breaks, even if you think you don't need them. Especially if you split sets.
8.) Don't underestimate the heat during evening/night matches. Humidity and hot air-temps can be just as dangerous as those things+the blazing sun. I recommend ALL of these tips (-the sunscreen) for night matches.
9.) Don't be a hero, unless you need to be a hero. Don't try to tough it out. It's not worth it. If you are cramping on the court, start to get really confused (forgetting scores constantly), or stop sweating, I strongly encourage you to stop play and get out of the sun immediately. The heat can be
deadly. Retiring/rescheduling/suspending a match is never deadly. If your opponent is being unsafe, and you become worried, offer to help (suspend play to run and grab a drink at the gas station, offer him an extra drink, hat, sunscreen, towel, whatever.) I've found that most will deny the help, but some won't. If your opponent becomes confused, or is obviously languishing, be prepared for him to collapse (hopefully this never happens, but if it does, get him into the shade and immediately call 911).
Hopefully my tone wasn't to serious, but I know people love tennis, and love to play, and that passion can often result in unsafe playing conditions, especially in super hot parts of the world (like Florida, Nevada, etc ...). It only takes you getting sick once before you never underestimate it again, and for some people, that "once" is a heat stroke and they die.
On a happy note--I get to play a match tomorrow! Yeah!