The function "LaplaceTransform" takes three arguments. The first argument is the expression to be transformed. The second is a symbol which indicates the independant variable of the first expression. The third is a symbol to be used in the output. If we take a look at your example,
LaplaceTransform[g''[x], t, s]
we see that the expression is g''[x] and the independant variable of it is "t". The expression g''[x] is a constant with respect to "t" and that's why you get that result. Ignoring the subtilies of analysis for a bit, the input:
LaplaceTransform[expr, t, s]
Is basically just:
Integrate[expr Exp[-s t], {t, 0, Infinity}]
If your expression "expr" doesn't have the variable of integration in it, then it's basically just a constant as far as the function "LaplaceTransform" is concerned.
If you are still seeing issues, try writing it out as an integral using the Integrate command. I think seeing that would help us better understand what you're looking to do.