Tidy up my bit string
Python 2, 127 103 101 bytes



lambda a:min(zip(*[[1,0,i%2,~-i%2]for i in range(len(a))]),key=lambda N:sum(o^b for o,b in zip(a,N)))


Try it online!

Makes the lists 111..., 000..., 101... and 010... with the first zip then finds the min of these lists with key function: sum of the xor of each element in the each possible output the input.