Audio Frequency White Noise generation using Arduino Mini Pro

I don't know that you need all that - just an Arduino and a piezo speaker and a pot if you want to play with the volume.

void generateNoise() {
    unsigned long int newr;
    unsigned char lobit;
    unsigned char b31, b29, b25, b24;
    b31 = (reg & (1L << 31)) >> 31;
    b29 = (reg & (1L << 29)) >> 29;
    b25 = (reg & (1L << 25)) >> 25;
    b24 = (reg & (1L << 24)) >> 24;
    lobit = b31 ^ b29 ^ b25 ^ b24;
    newr = (reg << 1) | lobit;
    reg = newr;
    digitalWrite (speakerPin, reg & 1);
    delayMicroseconds (50);  // Changing this value changes the frequency.
}

Mike's answer uses a Fibonacci Linear Feedback Shift Register configuration, but the compactness Galois Linear Feedback Shift Register Configuration could be faster:

#define speakerPin 8

unsigned long lastClick;

void setup() {
  // put your setup code here, to run once:
   pinMode(speakerPin,OUTPUT);
   lastClick = micros();   
}


/* initialize with any 32 bit non-zero  unsigned long value. */
#define LFSR_INIT  0xfeedfaceUL
/* Choose bits 32, 30, 26, 24 from  http://arduino.stackexchange.com/a/6725/6628
 *  or 32, 22, 2, 1 from 
 *  http://www.xilinx.com/support/documentation/application_notes/xapp052.pdf
 *  or bits 32, 16, 3,2  or 0x80010006UL per http://users.ece.cmu.edu/~koopman/lfsr/index.html 
 *  and http://users.ece.cmu.edu/~koopman/lfsr/32.dat.gz
 */  
#define LFSR_MASK  ((unsigned long)( 1UL<<31 | 1UL <<15 | 1UL <<2 | 1UL <<1  ))

unsigned int generateNoise(){ 
  // See https://en.wikipedia.org/wiki/Linear_feedback_shift_register#Galois_LFSRs
   static unsigned long int lfsr = LFSR_INIT;  /* 32 bit init, nonzero */
   /* If the output bit is 1, apply toggle mask.
                                    * The value has 1 at bits corresponding
                                    * to taps, 0 elsewhere. */

   if(lfsr & 1) { lfsr =  (lfsr >>1) ^ LFSR_MASK ; return(1);}
   else         { lfsr >>= 1;                      return(0);}
}


void loop() {
      /* ... */
      if ((micros() - lastClick) > 50 ) { // Changing this value changes the frequency.
        lastClick = micros();
        digitalWrite (speakerPin, generateNoise());
      }

}