What are some "essentials" that you'd recommend every Sailor having with them prior to deployment?

Not trying to be a smart ass but, you deploy you should have a will.

ENS (Join to see) - Oral and barrier contraceptives. Whether or not troops are supposed to fraternize - young people (and not a few older people) occasionally get too close to each other - particularly when stranded on a moving island hundreds to thousands of miles from home. A supply of clean comfortable easy to maintain underwear. Preferred tampons and monthly period medications. Cell phone that can be used as a camera, wifi device, and portable music player. Headphones and extra headphones. Sunscreen 60+SPF. Tooth brush, hair brushes, scrunchies, skin care, razors, shaving cream, antiperspirant, and makeup supplies. Social skills, self defense training, nonlethal self defense devices, situational awareness, thick skin, and a sense of humor. Reading material - perhaps on an iPAD. Weapons where appropriate to your billet - though CMDCM Treants has suggested to me in the past that with rare exception lethal weapons should be secured in master at arms locker. I personally prefer to bring along some small binoculars - and a Nikon D810 with 2X extender and 24-70mm and 70-200mm VR f/2.8 telephoto lenses - together with small pocket Munsell color chart and lens cleaning materials. Some items may not be allowed on nuclear unterseebooten. Warmest Regards, Sandy :)
CMDCM Gene Treants SCPO (Join to see) Capt Tom Brown LTJG (Join to see)

I have been on 4 deployments 3 on ship and one as a Dirt Sailor. My last ship was a CG (Small boy) So here is what I have learned, and please keep in mind some of this is allowed by some commands and others don't.

TP is a Must 2 rolls min, For some reason the Military never seems to order it or if they do they don't want to hand it out.
A Sleeping bag (this can be over ridden by the command) make sure it can be folded and stowed at the foot of your rack, and don't buy an expensive one. Your going to throw it away when you leave your ship if not sooner. Every thing takes on the Boat smell. If you have been there you know what I'm talking about.
Uniforms ... (sigh) im not saying don't take everything but I am saying be sensible about it. Take 2 weeks worth of T-shirts, underwear and socks, you should be issued 2 sets of FRV's (coveralls) Take your dress uniforms and have them hanger inspection ready. If you are going to be standing a lot of QD watch's take 2 sets, one for Watch and one for Inspections. In fact you should have one of each uniform that is Inspection ready and never worn.
Others have said it 2 pairs of boots one for working the other for inspections and or emergency use.
As for stuff in the head I found a Microfiber towel was great it dried fast and never smelled. They come in really large sizes too. Personal entertainment I leave up to you just think small and light. You don't have a lot of space and you may or may not have a plug to recharge stuff. Amazon is your friend BUT plan for it to take a month or two to get to you and don't buy anything that is overly expensive in case it gets dropped overboard during the Resupply at Sea. As for personal clothing don't pack like your going on vacation. Maybe 3 to 5 nice shirts a pair of jeans and a pair of slacks work, anything more is dumb your not hitting the town every night and the locales don't care if you wear the same stuff a month later when you might stop their again. Your not out there looking for a Wife (if you are you have issues)

If you are a Supervisor / LPO you will have more space in your shop here is how I used that.

I had 2 plastic boxes (the kids in the shop called them the Magic boxes) that fit nicely under my desk. Didn't leave much leg room but it was ok. In one I kept ... how should I put this... Odd tools. things the Navy doesn't keep on hand or you may need. As an IT I had barrel connectors special cross over cables odd pinned connectors I knew we used on the ship some specialty tools ect. When your out at sea you don't have time to order and wait for it to arrive so plan ahead and accordingly. Look over your gear for that weird old item that is going to give your problems and plan for it.

The OTHER box.
Before we pulled out I found out what each person in my shop liked, one was sour patch kids another was Swedish fish, Chocolate ect ect. I packed that box (discreetly when no one was looking lol) full of those items and a few for my self lol. About 2 months into deployment the Ship store will be out of all the normal stuff and you get the weird foreign knock offs if that. When I would see a person in my shop just having the worst day possible or they are having deployment blues I break out that one thing they like and toss it at them. Makes their day gives them a moment of not the Navy and the ship. This can be that thing that helps bring them back in focus that it really isn't that bad, and build some Morale. I learned this from watching an old Senior Chief on my first deployment.

I think others covered a lot of the basics I wont duplicate here, I would also say talk to those that have deployed on your ship before. Find out what tricks and tips they have, every Ship has its quirks.