How do I create a very long delay with Redstone?

Solution 1:

Minecraftaddict's "Extreme Delay Redstone Timer" is the longest delay circuit I know of. (But I can see from the comments that you've already found it). However, directly using this circuit won't work with a light sensor output, since the output of a light sensor would be 10 minutes on and 10 minutes off.

The solution to this is to connect the output of your light sensor to a falling-edge and rising-edge monostable circuit that are OR'd together.

This is an image of a rising-edge trigger on the left, and a falling-edge trigger on the right. You would connect the A inputs together – to the output of your light trigger. When the A input is turned on, the Q output of the left circuit pulses. When the A output turns off, the Q output of the right circuit pulses. Because of this, you can connect the two Q output wires together (ORing them) to have an output that sends a signal whenever the light-sensor changes.

However, these circuits are designed to make a very short pulse. Because Minecraftaddict's delay circuit comes with a built-in pulse shortener, you will want to send a slightly longer signal to it. To do this, just add more repeaters to the edge triggers (add an extra repeater to when the current repeaters are). Leave the repeaters that should be on the first setting like that, but change the repeaters that should be on the second setting (according to the image) to the fourth setting.

Solution 2:

The longest I can think out of top of my head?

Fill the dropper with as many items as many days you want the signal to take. Near each noon the dropper will discard one item, for a max of 576 in-game days. If that's not enough, you can restock the dropper with a chain of hoppers and a double chests, and only when all the items are exhausted the signal will be produced.

If you want this retriggerable, with limit of 320 days you can "discard" the items into a hopper blocked with a piece of redstone that will be deactivated once the time is past, and feed all the items back in.

If you want something faster, clock that dropper with a pair of repeaters or any of hundreds minor clocks. With an adjustable clock and a specific number of items you will easily fit within that 3-5 minutes range.