Push and hold on/off switch for arduino
I would go for option 2.

You can wake up the controller immediately, but only the processor.

Then just start waiting for the "long enough" time that you defined, regulary checking the button state. If the button is still pushed, then initiate full wake up. If it is released, go back to sleep.

This is the easy option, requiring no extra hardware, and only a dozen of so of lines of code.

Another option, that exist and have a somewhat similar behaviour, is to have your processor wake up regulary on timer, and checking the state of the input. This is what is done for "radio controled" devices, for example (Where the "button" is a radio circuit that need to be switch on and off).
Given a sufficiently long sleep between check, on average you will get the "long enough" delay.

This option is useful when you don't have available interrupt pins, or you need something more complex that a button as the input.