The code I end up with is Program Exercise 8.2. Yours might be even better. I have left all the original code still there (although I will probably have to remove some of it later). When the button is held down the message changes and the counter updates.

I have managed to get the display to count up a value of 2 by quickly pressing and releasing it, but my experiments have convinced me that I have enough time resolution to make the program work. (in fact there is almost a game in seeing how small you can make the number - but this would result in a broken switch after a little while). If I hold down the switch I notice that the counter reaches 200 in a second, so the interrupt rate is correct.

/*  Ex 8.2 Reaction Timer         */
/*  David Miles April 2006        */

#include <system.h>

#include <lcdlib.h>

void setup_hardware (void)
{
    /* Set PortA to use digital inputs */
    ANSELA = 0x00;

    /* bits 0-4 of PORTA for input  */
    TRISA = 0x1f ;

    /* Configure the INTCON register  */
    /* to enable timer interrupts     */

    INTCON = 0b10100000;

    /* Configure the timer to use the prescaler */
    OPTION_REG = 0b01000100 ;
}

int time_count = 0 ;

void tmrHandler( void )
{
  /*  increment the timer if the  */
  /*  button is held down         */

  if ( PORTA & 0x01 ) {
    time_count = time_count + 1 ;
  }
}

/* This is the interrupt handler  */
/* It is called when the PICmicro */
/* detects an interrupt           */
/* It checks the status bits to   */
/* find out who caused the        */
/* interrupt and then calls that  */
/* handler                        */

unsigned char counter = 0;

void interrupt( void )
{
    /* if the timer has overflowed */
    /* bit 2 of INTCON is set high */

    if( INTCON & 4 )
    {
      /* clear the bit to turn   */
      /* off this interrupt      */
      clear_bit( INTCON, 2 );

      counter = counter + 1;
      if ( counter > 2 )
      {
        counter = 0;

        /* call the handler        */
        /* function                */
        tmrHandler();
      }
    }
}

void display_value ( int value )
{
  unsigned char hunds, tens, units ;

  /*  first get the digits  */

  units = value % 10 ;
  value = value / 10 ;
  tens = value % 10 ;
  value = value / 10 ;
  hunds = value % 10 ;

  /*  now display them      */

  lcd_print_ch ( '0' + hunds ) ;
  lcd_print_ch ( '0' + tens ) ;
  lcd_print_ch ( '0' + units ) ;
}

const unsigned char down [5] =
{
  'd','o','w','n',0x00
} ;

const unsigned char up [5] =
{
'u','p',' ',' ',0x00
} ;

void main ( void )
{
  lcd_start () ;

  setup_hardware () ;

  while (1)
  {
    lcd_cursor ( 0, 0 ) ;

    if ( PORTA & 0x01 )
    {
      lcd_print ( down ) ;
    }
    else
    {
      lcd_print ( up ) ;
    }

    display_value ( time_count ) ;
  }
}