What is the value of a 1866 US coin with a wreath and roman number III in the middle of the back and lady liberty on front stamped with initials HWV?

That coin is a three-cent nickel (there is also a smaller coin which was also worth 3 cents but struck in silver and is generally called a three-cent silver). Depending on condition your coin can be worth anywhere from about a dollar to about 5 dollars if it is circulated.