5 watts? If it's rated as 5W @ 5V, then it will draw 1A when supplied 5V, or 2A when supplied 10V, by ohm's law. That's not low power at all.
5V power supply + 555 timer + 1A NPN transistor on output
I was going to suggest a low power CMOS 555, but if you are drawing so much current from this supply for your solenoid, it's probably a miniscule proportion of the total current drawn either way. Still, I spose 1A of current for 10ms isn't too bad if you aren't triggering it frequently. Then you could greatly reduce reduce the standby current draw to less than 1mA.