I took a look at mini-ITX mobos but am lost at what to pick, including the CPU and the PSU to get a silent box. Any idea?
Just one: the allotted budget, how much is it?
Anyway, the Fractal should work cooler (and possibly quieter) than the more cramped Coolcube.
If you don't need an external GPU you can add a modular high efficiency PSU, like the Corsair RM450 or the BeQuiet E9, both a total overkill but there isn't about anything in the market around 200-250W.
With dual cores CPUs, you may safely go for the relatively inexpensive Silverstone SST-ST30SF SFX unit, which is not that efficient, not modular, but very small and semi-fanless up to 55°C, or with a classical all-arounder like the Seasonic G-360, usually fairly quiet under 100W DC.
Alternatively you may look for some PicoPSU setup, but maybe with different enclosures.
The hw could be fairly standard and not that expensive to run in the Node 304: an ASUS H87/H97, a base Core i5 (to be fair a Core i3 would suffice), a Scythe Kotetsu heatsink, a couple of 4/8Gb low voltage DDR3 sticks (G.Skill, Crucial), and a 512Gb Crucial MX100 or similar SSD.
Whether the Fractal stock fans should be too noticeable for your taste, some SPCR recommended fans may help you.