For mud runs, less is more. Try to get light, tight fitting clothes. Avoid cotton and wool as they pick up water. Any clothing items that wick away water are a good choice. Your clothes might feel light and comfortable dry but wet it can be a whole other matter. A large t-shirt might end up hanging down to your knees after running with it wet for several miles.
Remember whatever extra weight you have on you, you must carry throughout the entire event. This includes the extra weight from water, so keep it light.
Just like with clothes, only take what you absolutely must. Most events have water provided and possibly a banana or other food, so you shouldn’t have to carry any food with you.
Carrying your car keys or wallet is just a disaster waiting to happen. Even in a waterproof container the chances of losing them are great and the chances of finding them once lost, are pretty slim. Most events have places to securely store your personal items, or you may leave them with a spectator if you know one there at the event.
Hats, gloves, goggles are not necessary and may slow you down. Most events have a list of recommendations on what to wear. Read them closely.
You are at the event to have fun. Many encourage participants to sport costumes. Wearing one may make the event more difficult to complete but adds to the reward factor and goes with the spirit of the event.