I would refinance to get a shorter term (30 years down to 15) and a lower interest rate and then use the savings to pay off my mortgage quicker.
I would not refinance to take money out of the "equity" in my home.
If fact, I wouldn't really try to refinance. I would ask my mortgage holder to modify my current mortgage. Every time you refinance you pay closing costs and reset the clock back to 15 or 30 years. Modifying a mortgage only changes the interest rate and subsequent payment. It does not change the duration of the loan and you can not opt to take money out in the transaction.
My mortgage is with a credit union. I just asked to modify the interest rate on my 15 year, 4.875% fixed mortgage, but keep the same payoff date. At no cost, my credit union decreased my mortgage interest rate to 3.625% with a savings of $340 on my monthly payment. To pay off my mortgage even earlier, I will add the monthly savings as additional principle payment each month.
This probably won't work if your mortgage was sold or serviced by a third party.