My mother in-law is considering giving her house to my husband (he would re-fi in his name only and remove her name from the title) for him to use as a rental property.
He is paying for it. Depending on how much the mortgage is compared to the value of the home this may be considered a straight purchase where the mortgage amount is your basis or it may be considered a gift where StMichael's answer is correct
Then the basic premise is false. One can't "give" what one doesn't own. If the house has no mortgage then the MIL may "give" the house. Depending on the value of the house, gift taxes might be due. Thereafter the son may take a home equity loan (it wouldn't be a re-fi as there is no "fi").
The MIL may "sell" the house to the son and then it is his (no gift involved) and the existing mortgage is paid out of proceeds at closing. The MIL may be subject to capital gains taxes but probably not (2 of 5 years rule).
There is never a zero cost basis (otherwise you can never take depreciation).