Dissolving gallstones with medicine relies entirely on the stone being made of cholesterol, because oral bile acids such as ursodeoxycholic acid only break down cholesterol-rich stones. So the question really asks which feature tells you a stone is NOT pure cholesterol.
Cholesterol stones do not contain calcium and therefore appear radiolucent, invisible on a plain radiograph. The model patient for medical therapy has small, radiolucent, cholesterol stones sitting in a gallbladder that still contracts normally with an open cystic duct, all of which let the dissolved bile acids reach and erode the stone.
A radio-opaque stone is the red flag. Its visibility on X-ray means it is calcified, that is, a pigment or mixed stone, and such stones simply will not melt away with bile acids. That makes radio-opacity the contraindication. Every other listed option, namely radiolucency, a well-functioning gallbladder and small stone size, is actually a point in favour of attempting dissolution.
\[\boxed{\text{Radio-opaque stones}}\]