As was mentioned above, once you write var=... then the expression has been evaluated and too late to do anything about it. If you still want to have the form of the expression be there without the Hold showing up. You can use Defer like this

   var = Defer[3 x^3 + 6 x^3 + a 6 x^3]
   (*3 x^3 + 6 x^3 + a 6 x^3*)

Now you can do var /. (6 x^3 -> 0) then to remove the defer, replace it with Identity

Maybe there is a way to pass it as characters.

I would think that using strings to Manipulate algebraic expressions is not the right way to go about these things and defeats the purpose of having symbolic computation.

In your second example var was evaluated in the first line. So, short of wrapping the definition of var with something like Hold as in

var = Hold[3 x^3 + 6 x^3 + a 6 x^3];

there is not much you can do since by the time it gets to your HoldForm in your second line is has already been evaluated to

9 x^3 + 6 a x^3