I commend the proposer of this question since it is relative to the architecture and design that would be the focus of this discussion. The main focus would of course include the purpose of the system in order to pick a specific component of the whole. It would be similar to asking what is the most important part of your body. Well, I would have to say my brain, others may say the heart. Given the pretext of this discussion, I would say the processor since it is the main focus of any system and with great detail of time developing it is sought after. For example, The JVM processor that is a virtual processor using memory to allocate registers and an architecture of design in microcoding the instructions to execute a single instruction: as do all processors that have microcoding to execute say "add" or "subtract" two variables. Without a processor the body is a paperweight and of course vis versa. Processors have several different flavors such as the Tile64, Itanium 2, and IJVM processors which are just a select few that I find interesting. The amount of potential that some of these processors posses are amazing and specific in purpose to the design and solution to a specific problem. Itanium 2 would be parallel computing mainly with plenty of additional functionality such as predication, speculation, and branch prediction. The rest of the computer is important too, don't incorrectly judge me in saying that the processor is all important but it is substantially important to its function in a Harvard architecture.

LOCATION, LOCATION, LOCATION. it's all on the motherboard. it's about the expandability of this setup. it's about rather this base (motherboard) is future proof or not. it's about the tech built onto the board - chipsets, overall quality, compatibility. it's about will everything you want to have for your computer fit on this board. it's like who wants to build a skyscraper on a piece land without concrete platform and other additional support? PSU comes in second, it's tricky to pick a good one and it's almost a different for every setup, everything are easier to pick out- the latest and greater on special plus popular, bam!

pugg2002 made a good point, the Mobo is really important - a poor layout on it can throw-off your whole build.

I personally think that throughput of storage devices (mainly the HDD) need improvement and are lacking today. SSD are still prohibitively expensive are their capacity is small. The conventional hard disk drive has a good reliability, good storage capacity, but the drives in the average desktop hasn't gotten faster at the rate of nearly anything else in the industry - only capacity has grown at a great rate.

On a new "dream" build today, my biggest concern would be Hard Drive throughput. Capacity isn't that important - you can add 500GB ext. drives all day. But while communication on your system's bus is moving Gigabytes per second, the hard drive is only handing you computer information at tens of Megabytes. I think, if you build a system based on the hard drive interconnect, your end result will be very satisfactory. Example: if you start with a SCSI or SAS controller, this gives you high performance drives. You then need a motherboard that is compatible, say a pci-x8 interface for the controller card (or the controller is onboard) this pretty much leaves you with a server board. As long as it has a good gfx card interface (pci-x16) your golden IMO. At that point RAM, CPU, etc., is limited to a selection that will surely be satisfactory, especially compared with a desktop at your local retailer. Oh yeah, and always get the drives and controller for a nice RAID array.

'course, I guess if your running linux and have a ton of ram, you could just make a nice ram drive and be done with it.