If you are set on the Prodigy (i like it also), i would first check
OCN Bitfenix Prodigy Owner's Club it has a ton of build there from simple stock cooling, to very complex water cooling, but really very impressive builds, the case is very flexible into what you can fit into it, that said is not that small, its close to the size of micro atx builds since it fits normal atx psu, 200mm fans, etc, but cerntanly a case that has its appeal.
As Pappnaas already said, T CPUs are just downclock versions, no special binning or low voltage, no cherry pick CPUs, i own a i3 2100T and really the only reason to pick them are its factory low profile cooler that's its smaller than the standard. I would recommend a 4770 or the upcoming 4771, at idle all the Haswell CPUs (as it was the same case in my experience with sandy bridge) downclock to the same idle clocks, 800mhz, this will be true for T series also, the only ones that i had my doubts were some Xeon L series, and were extremely expensive CPUs, only OEM for server builders, i almost bought one on ebay.... but almost $300 for dual core, and i had a good offer on my xeon 1230 so i passed. But dont go with T, go with the standard it should be fine, specially if you go with good tower cooler, i have my 4770K at stock clocks and it idles at 30C on 650rpm noiseblocker PLPS, and even under load i reach 70C on 850rpm, and this is 120mm fan, i bet a better fan will allow even lower temps.
For CPU cooler, i would avoid water cooling (unless you willing to spend lots of money), i would ask in the thread above if you can do passive watercooling, personally i doubt it, the water will slowly go up and up, and with no direct airflow on them it will overheat. In most scenearios air will end up quieter, there are very good air coolers available now days. If you can wait some for Scythe USA to reopen, i would try to wait for Scythe Mugen 4, was recently reviewed by SPCR and got the editor choice, i own Scythe Mugen 3 that imo very similar, and great cooler, great fan, if you motherboard can control PWM fans it will be able drop the fan around 400rpm where it will virtually inaudible at idle, and even under load should be very quiet. If you wish to read more check
SPCR Scythe Mugen 4 CPU Cooler: Scythe Strikes Back. As a second recommendation, i would go with
Noctua NH-U14S, the NF-A15 PWM is a very good fan IMO, it can be drop down to 240rpm with fanXpert on CPU_Fan header, the only downside is much more expensive than Mugen 4, but its available right now. As a third recommendation check
Thermalright HR-02 Macho Quiet/Fanless Cooler, the issue here is that im not sure it will 100% fit with the back fan because of its asymmetrical design growing flush from the CPU to the back of the case, i think it will fit without the back fan that should be fine also, but will limit what you can test, etc, for this i would go with the Mugen 4 or Noctua NH-U14S. Twin tower coolers are also an option, check the thread above you will find tons of Phantek PH14, Noctua NH-D14, Silver Arrows, etc.
Now for motherboard, im going to recommend to go with ASUS, FanXpert imo is too good to pass up, it will give you great control over pwm fans on CPU_header and very good control on 3pin fans on voltage controled headers like CHA_FAN 1/2. The second reason and very important for Prodigy build for fitting a big Tower cooler is the postion of the CPU socket, on the Asus is a little more centered than the usual boards (asrock also has more oposite orientation to the PCIe socket). On Haswell there are 4 options, check them out see what fits your budget and connection needs,
link, both the ASUS H87I-PLUS and ASUS H81I-PLUS, seem very nice cheap options if you are not overclocking, on a side note they seem to have FanXpert but dont specify if its 2 or the old one, regardless either seems like a very good choice.
For memory, specially for a mini ITX builds, i would suggest to go with very low profile and low voltage,
Crucial Ballistix Sport 16GB (2 x 8GB) 240-Pin DDR3 SDRAM DDR3 1600 (PC3 12800) Low Profile Desktop Memory Model BLS2K8G3D1609ES2LX0 is what i would recommend (
cheaper atm on amazon), this would even allow you to upgrade to a twin tower cooler later on, the memory is working great on my GENE VI, but its not on QVList, so its you call, but i had a good experience with it, so i continue to recommend it. There is cheaper 8gb version if you dont need that much memory,
Crucial Ballistix Sport Very Low Profile 8GB Kit (4GBx2) DDR3-1600 1.35V UDIMM 240-Pin Memory Modules BLS2C4G3D1609ES2LX0For PSU i think you should be fine with picoPSU if thats what you want,
picoPSU-160-XT + 192W Adapter Power Kit is what i would recommend, but there might be an issue with the memory as in all the Haswell mini ITX mobos that i seen the 24pin is right in front of the memory, even with ultra low profile memory the caps on the front might touch or even prevent it to be installed, this is something you need to research, if you trully want to go with picoPSU, then i would probably go with i7 3770 +
Intel DH77DF, but the cpu socket is right next to PCIe slot so this will limit a lot of what coolers you can use, and intel hasnt released mini ITX Haswell motherboards, maybe when they release the dual cores they might. For the before, i would also consider a standard ATX PSU, specially since the Prodigy can fit it, consider
SeaSonic SSR-360GP, reviewed by
SPCR Seasonic G360 PSU: High efficiency & performance, low price, kinda ideal for the Prodigy in terms that its always on sucking from the bottom, the Watts are more than enough for it, its cheap and its smaller than the 160mm standard (i think its 140mm) so it should fit very easily on the bottom chamber (some 160/170 do have a harder time to fit and with the cables).
When i was considering doing a Prodigy build, i had the
BitFenix Spectre Pro 200mm Fan - All White BFF-SPRO-20025WW-RP on my list, read some decent comments on it, i bet with fanXpert it can be brought down to inaudible levels and great to cool all the case, maybe even fansless on the cpu cooler, this is more to test upon building.
If you build it, try to post some pics, i always like Prodigy builds, and your experience in terms of noise and temps. Good luck with the build.