My Mom tells me the same story over and over within a 5 minute period. What do I do?

Let it go! There really isn't anything you can do about it. Just be polite and listen (trust me, I know at times that's not easy) and remember in her mind its the first time she told you. The repetition is hard on family for sure. My Mom has told the same stories of the old days over and over, but I've noticed in the last few months as she progresses down the dementia road her memory of those stories of old is fading. She has vague recollection of the details now and
the retelling of those stories is less frequent. Its really rather sad, because those memories were a major part of her.
When I talk with Mom anymore I just accept the fact that she won't remember what we've talked about. We still enjoy each others company but in a different way. Conversation for the sake of conversation isn't a big priority anymore. Once I accepted the fact that Mom couldn't remember what we talked about 5 minutes ago things became easier for me.

Definitely let it go. In order to not lose YOUR mind, come up with a different answer. Possibly silly ones? Once in a while they will "get it" that your being silly and get a chuckle. If your committed to helping her you need to find ways for her to enjoy this terribly confusing time for her.
I found directing the conversation in a different vein can help. Distraction.
I had an ah ha moment when I realized it was sure a lot like raising my 2 yr old.
Have fun with it. The older memories are strongest. Talk about her life and her growing up.

Let it go. It's not important and she can't help it. Yes it's exasterbating, and tiring...but just take a deep breadth and listen one mor time...then distract her by leading her to another room or starting a new activity. Took me a darn long time to get to the "let it go point" but now my outlook and interaction with mom is a whole lot better and she is less anxious...who wants to "repeat themselves" unknowingly?

Mom has Alzheimers and she now getting a little nasty. If I tell her we have already discussed it and I really don't want to talk about it any more, she hangs up on me. I know that is the wrong thing to do. I'm going to try changing the subject if I can. It doesn't matter, she won't remember what I say, or that she got mad, or I got mad because she will start over again on it tomorrow on the same subject as if it never happened, and for her, it hasn't.

That is the sad thing about dementia. They simply do no remember what you or they have said after about five minutes. Talking to my mother is like living in a tv show that loops around and starts over again and again. She can't help it, that is the way her mind works. There are only a few subjects she thinks about and she continues to go over and over and over them and obsesses over things. I can respond and a few minutes later, or the next day, its as if it is the first time she brought up the subject. I pray to God I am not going to be like that. It is really hard on the family, but there is no medicine that I know of that will stop it.

I remember my Dad use to complain about his own father who would tell the same stories on a regular basis.... interesting, my Dad has been doing the same thing the past couple of years... and the stories are almost word for word. I would just let it go.

Next time Dad will start with the same story about his college days, he has many, for the 100th time, I need to remember to ask him things I hadn't heard before, like more about the college dorm, the physics lab, his professors, etc.

With my dad, we tried to react as though it was the first time he'd mentioned it - in the early stages, if he sensed from us that he was repeating ( and forgot), it would frustrate and bother him. I see my mom's short term memory slipped and causing her to repeat - rolling with it seems to be the best way for me to deal with it. And yes, it can get wearing - neuro illnesses stink.

Almost everyone who is around someone with dementia has the same problem with repetition. My mother remembers maybe 20-30 events in her entire life. I have heard these events so many times now that I can quote them word for word. Strange thing is that her memory on telling the events is very good. She uses the same words each time. I don't say anything usually. Sometimes my veneer cracks and I say something cross. That hurts her feelings, though, so I always regret doing it. She is trying to communicate and doesn't mean to irritate me. I figure there will be a day when I wish I could hear the stories one more time. Until that day... the repetition sure can be crazy-making.

LET IT GO. Find an activity she can enjoy. Simple puzzles; looking at the pictures in a high style fashion magazine, folding towels or simple clothing items, etc.

You can't reason with someone who has lost their ability to reason. Even if you convince her of something she said or did a minute ago in the next minute it will be forgotten. You can find tons of info on dealing with dementia on this site. I'm dealing with the exact situation with my dad.