In Mathematica, everything is an expression. So, the expression fun1 is effectively the "function pointer" that you want. It's an expression with down values that specify how to evaluate when applied to arguments. Let's make this concrete:

fun1[x_Integer] := ToString[StringForm["fun1 accepts `` and doubles: ``", x, 2 x]];
fun2[x_Integer] := ToString[StringForm["fun2 accepts `` and squares: ``", x, x^2]];
fun3[x_Integer] := ToString[StringForm["fun3 accepts `` and factors: ``", x, FactorInteger[x]]];

Let's simulate the user's choice with a random choice:

chosenFunc = RandomChoice[{fun1, fun2, fun3}];

Now let's use our choice:

chosenFunc[18]
(* "fun2 accepts 18 and squares: 324" *)

I think this is cleaner:

chosenFun =
 Switch[
  param,
  1, fun1,
  2, fun2,
  3, fun3,
  _, Null]

You would do something more appropriate than just Null in the fall-through case. If, Switch, and Which are special forms that hold their arguments unevaluated, so you don't need to worry about unexpected evaluation as a side-effect.

But, another way to view this is as a parameterization to your function, like Hans suggested. An alternate way to do that would be:

newFun[1] = fun1;
newFun[2] = fun2;
newFun[3] = fun3;

And now you can just use:

newFun[param][x]