Using a NIC to do Wake on LAN? Or alternatives
Most NICs will need configuration through a PCIe or PCI host before they will be wakeable-on-LAN, so I have some practical doubts on your approach.

Also, Wake-On-LAN isn't based on unicorn magic, but electricity. It works by letting the ethernet controller in the NIC run continuously but not do much stuff, just listen for a packet containing a certain byte sequence. Hence, instead of powering your probably low-energy optimized device, you're trying to run PC hardware which will itself need a power supply. In a PC, there's a few supply chains running even when the PC is off, exactly to support this kind of operation. By the way, WoL also means that your ethernet switch needs to be forwarding broadcast packets to the (sleeping) computer, which means the ethernet port can't be switched off – which wastes power again. The question whether letting your single board computer run continously is so much worse than letting a ready-to-wake-up circuitry run can only be confirmed by power consumption measurements of your system.

I'd strongly suggest looking into your board (which you sadly don't specify) and its low-power modes. How long does it take to go from suspend-to-ram to running? How much power does it consume in run and in sleep mode? Is it feasible to wake it up for a split second every 20s, send a broadcast packet that it's up, wait a couple milliseconds for a reply packet, and if nothing signals that there's reason to stay awake, goes back to sleep. That way, the computer only needs a small fraction of its run-mode power, still is available in acceptable time when you need it, and also, the network switch can power down its ethernet port – which might actually compete with your computers "not doing much at all" power consumption!