Help with shrinking program on to Tiny85

That's excellent news.

Confirming the changes my side were removing the Serial.begin() and these pin assignments

int switchPin = 3;              // IC leg 2 (PB3) 
int led1Pin = 0;                // IC leg 5 (PB0)
int led2Pin = 1;                // IC leg 6 (PB1)
int led3Pin = 4;                // IC leg 3 (PB4)

Also, here's how it would look modified to use the internal pull-up resistor on the switch (as mentioned earlier). For this circuit you'd not have any additional resistors on the switch circuit, just wire to the switch and from there to ground.

/*
 *  Night Lite, final version
 */

int switchPin = 3;              // IC leg 2 (PB3) 
int led1Pin = 0;                // IC leg 5 (PB0)
int led2Pin = 1;                // IC leg 6 (PB1)
int led3Pin = 4;                // IC leg 3 (PB4)

int val;                        // variable for reading the pin status
int val2;                       // variable for reading the delayed status
int buttonState;                // variable to hold the button state

int lightMode = 0;              // What mode is the light in?

void setup() {
  pinMode(switchPin, INPUT);    // Set the switch pin as input
  digitalWrite(switchPin, HIGH);  // set internal pull-up resistor
  pinMode(led1Pin, OUTPUT);
  pinMode(led2Pin, OUTPUT);
  pinMode(led3Pin, OUTPUT);


  buttonState = digitalRead(switchPin);   // read the initial state
}

void loop(){
  val = digitalRead(switchPin);      // read input value and store it in val
  delay(10);                         // 10 milliseconds is a good amount of time
  val2 = digitalRead(switchPin);     // read the input again to check for bounces
  if (val == val2) {                 // make sure we got 2 consistant readings!
    if (val != buttonState) {          // the button state has changed!
      if (val == LOW) {                // check if the button is pressed
        if (lightMode == 0) {          // if its off
          lightMode = 1;               // turn lights on!
        } 
        else {
          if (lightMode == 1) {        // if its all-on
            lightMode = 2;             // make it blink!
          } 
          else {
            if (lightMode == 2) {      // if its blinking
              lightMode = 3;           // make it wave!
            } 
            else {
              if (lightMode == 3) { //  if its waving, 
                lightMode = 4; 
              }
              else{
                if (lightMode == 4){
                  lightMode =0;
                } // turn light off!
              }
            }
          }
        }
      }
      buttonState = val;                 // save the new state in our variable
    }

    // Now do whatever the lightMode indicates
    if (lightMode == 0) { // all-off
      digitalWrite(led1Pin, LOW);
      digitalWrite(led2Pin, LOW);
      digitalWrite(led3Pin, LOW);

    }

    if (lightMode == 1) { // Blue On
      digitalWrite(led1Pin, HIGH);
      digitalWrite(led2Pin, LOW);
      digitalWrite(led3Pin, LOW);
    }

    if (lightMode == 2) { // Blue off, Green On
      digitalWrite(led1Pin, LOW);
      digitalWrite(led2Pin, HIGH);
      digitalWrite(led1Pin, LOW);

    }
    if (lightMode == 3)  { // Red On Green Off
      digitalWrite(led1Pin, LOW);
      digitalWrite(led2Pin, LOW);
      digitalWrite(led3Pin, HIGH);
    }   
    if (lightMode == 4) { // Flame
      analogWrite(led1Pin, random(120)+135);
      analogWrite(led2Pin, random(120)+135);
      analogWrite(led3Pin, random(120)+135);
      delay(random(100));

    } 
  }
}

Cheers ! Geoff