When an elderly asks the same questions repeatedly, is it best to respond each time as if it's the first time it's been asked, or let them know they already asked the question?

Telling her she already asked will not help her remember that she already asked and will more likely than not bring some kind of conflict - further confusion, sadness/shame, more questions, argument, etc. Try to keep a positive tone and not let the frustration you're experiencing show while you answer again. If the same question is coming up over and over with no end in sight try a distraction, suggest doing something else or bring up a new topic for her to focus her mind on. When my dad keeps repeating the same question I always, for some reason, think that once he hears my same answer for the 2nd, 3rd, 4th.. time he will remember that he has already just heard it, but that never happens. My goal in that situation is to try and avoid pointing anything 'wrong' that would cause him to feel 'dumb' or ashamed while maintaining a level of my own sanity.