Try setting the GenerateConditions option to True to find a conditional expression where the results are valid.
Convolve[x DiracDelta[x], g[x] , x, y,
GenerateConditions -> True] (* 0 *)
Convolve[g[x], x DiracDelta[x], x, y,
GenerateConditions -> True] (* 0 *)
It appears that the answer is "0" for all conditions and for any function, g[x].