The type of mutation that can result in a new function for a protein is a missense mutation. This mutation occurs when a single nucleotide change in the DNA sequence leads to the substitution of a different amino acid in the resulting protein. Depending on the specific amino acid change, this can alter the protein's structure and potentially give it a new or enhanced function. Such mutations play a critical role in evolution, as they may provide beneficial traits that can be subject to natural selection over time.