You don't need an external timer if you have a microcontroller. You can use a Real time clock with any arduino. And I would use a mosfet as the switch and not a relay.
You can use a solid state relay rated for your application. I don't know if the control voltage can be driven by arduino or you might have to still use a mosfet for that.
Well if you want 1 second interval you'll need to make your own.
Commercial and hence "cheap-as" timers are made to suit the masses, not an individual requirement.
uC, real time clock, relays or mosfets output, power supply, plus programming and enclosure with display readout etc. etc. all add up to make an expensive item in comparison.