When looking at always-on devices, it rarely makes sense to re-use a device originally targeted at different use cases - unless very careful consideration and focusing has gone into the original scoping for low idle power consumption. For devices running 24/7 even small'ish deltas in power consumption can make a huge difference and basically pay for a new purpose-built device in a year or less (just on electricity and/ or cooling costs saved). E.g., if a device comes with a discrete graphics card, it's basically out of the consideration (because that means at least +20 watts idle power consumption, much more for gamer graphics).
If an Intel system is pre-haswell (11-15 watts idle possible with haswell or newer), you're looking >35 watts idle for ivy-bridge or >60 watts idle for sandy-bridge - and 110-130 watts idle for anything before that. For Intel atom based systems you want at least baytrail-d (and you don't really want baytrail-d either, because of serious silicon bugs affecting ACPI and deep c-states - but at least they give you low idle power consumption (6 watts) and considerable performance).
For AMD, the situation is a bit more difficult - the old APUs are a bit long in the tooth (and not as low power as comparable Intel systems), for the ryzen generation the corresponding low-power APUs haven't really arrived yet.
Unless you really need the power most of the time, you don't want (real-) octo-cores (plus HT) either, as Intel has rushed them to market in shock, because they didn't have anything to compete with ryzen on the high-end, with idle power being irrelevant during the rushed development.
 The pcengines apu2 is a bit of an outlier here, yes, it isn't a performance monster either and can just barely do 1 GBit/s routing at wirespeed under linux (not quite, but almost under FreeBSD) - and certainly not at all with SQM enabled or, beware, running a VPN, nor does it provide any kind of graphics output, but it does provide >2 (three) ethernet interface plus the ability to connect two WLAN cards in a small form factor and low idle power consumption.