Catey Hill of the Wall Street Journal advises against refinancing your mortgage if you're over 50, since "carrying a mortgage into retirement has traditionally been considered a bad idea." But if you do, get a 15-year mortgage instead of a traditional 30-year mortgage:
For starters, a 15-year term leaves more money in your pocket — you'll pay much less in interest over the life of the loan. If you refinance a mortgage (assume a 5% rate) balance of $100,000 for a 30-year term, you will pay more than double what you'd pay had you opted for a 15-year term (about $93,000 in interest vs. about $42,000). Plus, you can get a historically low rate on a 15-year loan now, too.
I'd go further than this. Say you're 55 and you have ten years left on your current mortgage. If you refinance the outstanding balance at a lower rate on a 15-year note, you don't have to pay it off in 15 years. You can pay it off faster by making the same monthly payments as you did with your old mortgage. If you do this, you'll end up paying off the new loan in, say, seven years with the same payments as before.
Obviously this depends on the exact terms of the loan, and it also depends on your ability to stick to your old payment schedule. If you know you're short on self discipline, this might not be a good idea. But if you're the kind of person who will keep making the payments religiously even though you could get away with paying less, it's a complete win.
UPDATE: As several commenters point out, you have to look at actual numbers to know if this makes sense. It depends a lot on how good a rate you get, what your old rate is, how many years are left on your current loan, etc. But in a lot of cases, I suspect that if you run the numbers you'll find that you can keep making the same monthly payments but pay off your outstanding balance faster. It's worth checking out.