User-changeable batteries or rechargeable batteries are the way to go. Some dive computers require it to be returned to the manufacturer for the battery to be changed which can take a couple weeks or more. This can be a big pain if you don't notice that you need a new battery until the day before a dive trip. The Mares Matrix is a good option for rechargeable watch style computer. The Mares Puck Pro and Suunto Zoop computers are both good user changeable battery computers.
How's your eye sight? Small watch style computers are stylish but can be hard to read. Some computers have screens that are easier to read than others. A couple that have really great screens are the Mares Nemo Wide and the Liquivision Lynx.
This is a great option to have included in your dive computer. If this is an option that you want in a computer, you will need to decide between hose mounted or wrist mounted with a transmitter. My preference is wrist mounted. When I'm in a good trim horizontal position with my hands clasped in front of me it's easy to view the screen. It's also nice to be able to take your computer with you to do your log book without having to take your whole reg. Some hose mounted dive computers do have a quick release option to help with this. A good wrist mounted computer with transmitter is the Liquivision Lynx. With this computer you can actually have multiple transmitters connected to it so you can keep track of your buddies air too.
Type of diving
Where do you see yourself going with your diving? Most basic dive computers will come with air and nitrox modes, but there's more. If you're attracted to the TEC courses you'll need to be able to switch between gases. Good computers for TEC 40, 45 & 50 are the Mares Nemo Wide, Liquivision Lynx, or the Mares Matrix. If sidemount diving is in your future then the Liquivision Lynx with it's ability to monitor multiple transmitters is a great choice. If you're attracted to the depths than trimix diving might be in your future. For trimix you'll need a computer such as the Liquivision X1.
As with all your gear, you'll want to think about service. You might find a really cool computer with lots of features for a great price. But if you don't have a local dive shop that can take care of service and warranty issues it won't turn out to be such a good deal in the end.
Of course your budget is going to strongly influence what computer you'll get. As with all your gear, I would suggest to get the best you can afford to save you from re-buying later. Although a basic computer can always function well later as a back up. Many divers will carry a back up computer on them. This way if your main computer fails you can switch over to your back up and not miss a dive!
Ultimately owning your own computer can make you a safer diver. Knowing how it works and how to read the information it's giving you is key. Get you're own computer, learn how to use it and go diving!