How do I create a 24 hour timer circuit capable of turning a servo on and off
Really, go for microcontrollers. 

Having a microprocessor with a stable external oscillator / quartz isn't expensive at all.

For example, take a sub-$1 ST STM32F030K6. If put into stop mode with an external oscillator for the integrated Real-time clock, it'll consume less than 10 µA, including leakage current in decoupling capacitors.

First, to put this into perspective: A typical AA battery has a self-discharge rate of 4% per year, at a capacity of 2.8 Ah (says German wikipedia). That means it has a continuous self-discharge current of ca. 12.8 µA!

In other words, if your AA battery was good for about 10 years lying around in a cupboard, it's now worth 5 years lying around in the cupboard, driving your microcontroller in stop mode. The fact that you'll turn your servo will cost many many times more energy than keeping the real-time clock running.

Regarding configuration:

yes, dedicating a few of the microcontroller's pins to a DIP switch will work to let the user select e.g. the time between turning back and forth; other options would include:


attaching a (debounced) button to an wake-up capable pin
a nice turn knob
pin headers and jumpers
software on a PC (you can buy similar microcontrollers that come with USB!)
a potentiometer to let the user smoothly select a time (be sure to only apply power to the potentiometer very rarely – otherwise, it'll be a significant drain of power, unless you choose a very large resistance, but that might be hard to properly read out)
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