Given the choice of taking 500mg of Naproxen once a day, or 250mg of Naproxen twice a day (separated by ~12 hours), which is safer for the kidneys and liver?
Naproxen has a half-life of between twelve and seventeen hours.1 Constant steady-state serum levels of Naproxen are produced after four to five days..1 It stands to reason that the lower the dose the lower the renal and hepatic load and vice versa; the higher dosage requires the kidneys and liver to remove more of the drug. @D Bagnall is correct, however, that the half-life is not affected by the dosage, other than the obvious fact that a higher initial dose leaves more drug in your body after the half-life period than a lower dose.
Most NSAIDs produce their effect by inhibiting the production of the COX-1 and COX-2 enzymes.2 COX-1, and (to a lesser effect) COX-2 help protect the stomach from its acidic environment.3 Contrary to popular believe, NSAIDs do not "burn" a hole in your stomach. Because of the protective effect of these enzymes, NSAIDs should be take only while symptoms persist. If needed for longer periods, you should consult with your physician.
But to answer your main question: As far as your liver and kidneys care it's the dosage that matters. One should always take the lowest dose that produces the desired effect.
Short answer: the kidneys and liver don't care if it's 500mg once a day or 250mg twice a day.
One must consider two things: the amount of medication needed to relieve symptoms and how long it lasts.
The body metabolizes 500mg and 250mg at the same rate, therefore 500mg won't last twice as long as 250mg. But if the 250mg isn't strong enough to reduce the symptoms, it makes no sense to take it at that dose.