Three mistakes,
1.	scanf("%s", &my_string); should be scanf("%s", my_string);
2.	char * new_string = (char *) malloc(sizeof(char) * (length + 1));
should be 
char * new_string = (char *) malloc(sizeof(char) * (length + 2));
3.	for (i = 0; i < length; i++)
should be
	for (i = 0; i < (length + 1); i++)
The correct version is the following code,
#include <stdio.h>
#include <stdlib.h>

int string_length(char * my_string)
{
  int counter = 0;
  while(1)
  {
    if (my_string[counter] != 0)
    {
      counter++;
    }
    else
    {
      break;
    }
  }

  return counter;
}

int main()
{
  char * my_string = (char *) malloc(sizeof(char) * 10);
  printf("please enter a string:\n");
  scanf("%s", my_string);

  int length = string_length(my_string);
  char * new_string = (char *) malloc(sizeof(char) * (length + 2));

  new_string[0] = '?';
  int i;
  for (i = 0; i < (length + 1); i++)
  {
    new_string[i+1] = my_string[i];
  }
  
  printf("new_string = %s\n", new_string);
  free(new_string);
}