The first option is obvious, one could hand build their own system. I do recommend buying a system that is built for the Elise/Exige so that it is plug and play. No cutting of the stock harness is a good thing.
HID stands for High Intensity Discharge, a type or arcing lamp. They require a ballast to maintain the arc. HIDs will pull a surge of current to strike the arc. HIDs will put out more light, and a cleaner or brighter light. Often noted is a bluish tint to the light, which depends on the color temp of the bulb. HIDs can be more bothersome to other motorists, so extra care in aiming is warranted. HID bulbs should last longer than standard Halogen bulbs. It is not recommended though to use HIDs for a bulb that is only lit for a short time as they take time to get up to full brightness. Flashing HID brights might not work as well.
The typical HID kit only upgrades the standard (non-bright) headlights. It is recommended that you look into an upgraded bulb for your brights and driving lights.
There are two main suppliers to the Elise-Exige market for HID kits. Elisetalk member Ekological has been supplying a custom kit for a while, and he supplies it with different color temp levels. Sector111 has a kit for sale. I will not try to go into the differences between the kits, that will be up to you to investigate.
Link to Ekologikals' kit $415 for 2005/2006 cars using Philips Apexcone 50watt burners. $340 for 2007 cars using Philips Apexcone 50watt H7 burners. Lumens are per the selected color temps (3000k to 8000k). Ballast is 50 watts. Lifetime warranty on all parts including the bulbs.
Link to Sector 111 kit $425.00 using Philip's bulbs that put out 3,200 lumens. Ballast is 35 watts. I believe this is a 3000k color temp kit.