24 AWG for 250 meters works out to about 20 Ohms. Using all four pairs should reduce that to approximately 5 Ohms.
16 watts at 12 volts means 16/12= 1.34 amps.
1.34 Amps through 5 Ohms means a 6.7 volt drop in the wire, and about 9 watts lost in the wire.
So, if you wanted to do this directly, you'd need to start with at least 18.7 volts and 25 watts at the source to meet your requirements at the target.
The losses in the wire depend on the current, not the voltage though. If at all possible, you'd be much better off transmitting higher voltage at lower current. For the obvious example, if you used power from the wall (240 volts) you could use a normal 12 V power supply at the far end.
Let's assume your power conversion at the far end is 80% efficient, so to get 16 watts out, you need 16/0.8 = 20 watts in. At 240 volts, 20 watts means about 84 mA. That gives a loss in the wire of roughly 0.4 volts, and 35 mW.
In short, in this case you don't really need to do much of anything to compensate the wire. Simply for safety, you might want to use an isolation transformer, and probably a fuse/circuit breaker specifically for this circuit. Given the requirements, something like 1/10th or 1/8th amp would be adequate.
Most Cat 5e cable I've seen is rated for around 300 or 350 volts1, so running 240 volts through it shouldn't be a real problem. Most is rated only for indoor use though, so using it outdoors might not work out so well (e.g., insulation might not withstand the UV in sunlight well, and might deteriorate fairly quickly). Ideally, you'd like it to be outdoor rated, but that may not be as inexpensive as what you're currently considering.
On the other hand, as @mkeith pointed out in a comment, carrying a high voltage on a cat 5 cable is sufficiently unexpected that it carries a degree of danger, even when/if the cable itself is rated for that voltage. It's probably safer to reduce the voltage somewhat, and put up with greater loss in the cable (though you might consider some middle ground, where you use a voltage somewhat higher than the 12 volts required by the router, but still lower than the 240 supplied from the socket--perhaps 48 volts, for example).
- For one example: UL Voltage Rating: 300 V RMS.