One note; in your code check out these lines:

i++;
if(i>=NUM_LEDS+NUM_LEDS) { i=0; }

In your case it will keep resetting the for-loop and make it keep going and going and going ...

Under ideal circumstances we'd want the leds to be addressed even when beyond number of LEDs.
What I mean with that; if we have 10 leds, and the counter (i) is goes to the 11th led, then that 11th led should become the first led.

void meteorRain(byte red, byte green, byte blue, byte meteorSize, byte meteorTrailDecay, boolean meteorRandomDecay, int SpeedDelay) {  
  int i=0;
  setAll(0,0,0);
  
  for(int i = 0; i < NUM_LEDS+NUM_LEDS; i++) {
    
    // fade brightness all LEDs one step
    for(int j=0; j < NUM_LEDS; j++) {
      if( (!meteorRandomDecay) || (random(10)>5) ) {  fadeToBlack(j, meteorTrailDecay); }
    }
    
    // draw meteor
    for(int j=0; j < meteorSize; j++) {
      if( ( i-j < NUM_LEDS) && (i-j >= 0) ) 
          { setPixel(i-j, red, green, blue); } 
      else // try removing this: if (i<j)
          { setPixel(NUM_LEDS-j, red, green, blue); } // added this so it does "negative" LED positions as well
    }
   
    showStrip();
    delay(SpeedDelay);
    i++;
    if(i>=NUM_LEDS+NUM_LEDS) { i=0; }
  }
}

In the "draw meteor" section, you may want to toy with that loop.

Yes, it is a computer case - the LED strip goes in a ring around the edge (that's why I want the effect to be a full circle instead of cutting off at the end of the strip). You can't see the strip in the first picture, but you can in the second.

Anyways, the code you posted this time did work - the effect the same as the code from the original post, I'm assuming this is what you were going for just to make sure everything works properly. 

For video I recommend pulling it through HandBrake to get a more efficient file size.

the link of the picture didn't work, and when I changed "http" to "https" I got a picture of PC case ... ?

Anyhoo ... we must be overlooking something simple, although I'm surprised that FastLED isn't working for you. Maybe the knock off Nano is to blame, but I kinda doubt that.
I still do not have my hardware available to test, so let's see what could be wrong with the code. I have moved around code a little to make it more readable for myself ...

This is what should work, or at least should show "something" ....

void meteorRain(byte red, byte green, byte blue, byte meteorSize, byte meteorTrailDecay, boolean meteorRandomDecay, int SpeedDelay) {  
  int i=0;
  setAll(0,0,0);
  
  while(true) { // was: for(int i = 0; i < NUM_LEDS+NUM_LEDS; i++) {
    
    // fade brightness all LEDs one step
    for(int j=0; j < NUM_LEDS; j++) {
      if( (!meteorRandomDecay) || (random(10)>5) ) {  fadeToBlack(j, meteorTrailDecay); }
    }
    
    // draw meteor
    for(int j=0; j < meteorSize; j++) {
      if( ( i-j < NUM_LEDS) && (i-j >= 0) ) 
          { setPixel(i-j, red, green, blue); } 
      else if (i<j)
          { setPixel(NUM_LEDS-j, red, green, blue); } // added this so it does "negative" LED positions as well
    }
   
    showStrip();
    delay(SpeedDelay);

    i++;
    if(i>=NUM_LEDS+NUM_LEDS) { i=0; }
  }
}

I'm not sure what I'm overlooking if this one doesn't work - then again, it's early in the morning here and I may need some more coffee 

#include <Adafruit_NeoPixel.h>
#define PIN 3
#define NUM_LEDS 81
// Parameter 1 = number of pixels in strip
// Parameter 2 = pin number (most are valid)
// Parameter 3 = pixel type flags, add together as needed:
//   NEO_KHZ800  800 KHz bitstream (most NeoPixel products w/WS2812 LEDs)
//   NEO_KHZ400  400 KHz (classic 'v1' (not v2) FLORA pixels, WS2811 drivers)
//   NEO_GRB     Pixels are wired for GRB bitstream (most NeoPixel products)
//   NEO_RGB     Pixels are wired for RGB bitstream (v1 FLORA pixels, not v2)
Adafruit_NeoPixel strip = Adafruit_NeoPixel(NUM_LEDS, PIN, NEO_GRB + NEO_KHZ800);

void setup() {
  strip.begin();
  strip.show(); // Initialize all pixels to 'off'
}

void loop() {
  meteorRain(0xff,0xff,0xff,10, 64, true, 30);
}

void meteorRain(byte red, byte green, byte blue, byte meteorSize, byte meteorTrailDecay, boolean meteorRandomDecay, int SpeedDelay) {  
  setAll(0,0,0);
  
  for(int i = 0; i < NUM_LEDS+NUM_LEDS; i++) {
    
    
    // fade brightness all LEDs one step
    for(int j=0; j<NUM_LEDS; j++) {
      if( (!meteorRandomDecay) || (random(10)>5) ) {
        fadeToBlack(j, meteorTrailDecay );        
      }
    }
    
    // draw meteor
    for(int j = 0; j < meteorSize; j++) {
      if( ( i-j <NUM_LEDS) && (i-j>=0) ) {
        setPixel(i-j, red, green, blue);
      } 
    }
   
    showStrip();
    delay(SpeedDelay);
  }
}

void fadeToBlack(int ledNo, byte fadeValue) {
 #ifdef ADAFRUIT_NEOPIXEL_H 
    // NeoPixel
    uint32_t oldColor;
    uint8_t r, g, b;
    int value;
    
    oldColor = strip.getPixelColor(ledNo);
    r = (oldColor & 0x00ff0000UL) >> 16;
    g = (oldColor & 0x0000ff00UL) >> 8;
    b = (oldColor & 0x000000ffUL);

    r=(r<=10)? 0 : (int) r-(r*fadeValue/256);
    g=(g<=10)? 0 : (int) g-(g*fadeValue/256);
    b=(b<=10)? 0 : (int) b-(b*fadeValue/256);
    
    strip.setPixelColor(ledNo, r,g,b);
 #endif
 #ifndef ADAFRUIT_NEOPIXEL_H
   // FastLED
   leds.fadeToBlackBy( fadeValue );
 #endif  
}

void showStrip() {
 #ifdef ADAFRUIT_NEOPIXEL_H 
   // NeoPixel
   strip.show();
 #endif
 #ifndef ADAFRUIT_NEOPIXEL_H
   // FastLED
   FastLED.show();
 #endif
}

void setPixel(int Pixel, byte red, byte green, byte blue) {
 #ifdef ADAFRUIT_NEOPIXEL_H 
   // NeoPixel
   strip.setPixelColor(Pixel, strip.Color(red, green, blue));
 #endif
 #ifndef ADAFRUIT_NEOPIXEL_H 
   // FastLED
   leds.r = red;
   leds.g = green;
   leds.b = blue;
 #endif
}

void setAll(byte red, byte green, byte blue) {
  for(int i = 0; i < NUM_LEDS; i++ ) {
    setPixel(i, red, green, blue); 
  }
  showStrip();
}

First of all; I'd recommend using FastLED instead of NeoPixel.
As for the code (still not able to test this myself), I think I may have found a boo-boo ... (3rd line from the bottom, added a "-1");

void meteorRain(byte red, byte green, byte blue, byte meteorSize, byte meteorTrailDecay, boolean meteorRandomDecay, int SpeedDelay) {  
  setAll(0,0,0);
  int i = 0;
  
  while(true){
    
    // fade brightness all LEDs one step
    for(int j=0; j<NUM_LEDS; j++) {
      if( (!meteorRandomDecay) || (random(10)>5) ) {
        fadeToBlack(j, meteorTrailDecay );        
      }
    }
    
    // draw meteor
    for(int j = 0; j < meteorSize; j++) {
      if( ( i-j <NUM_LEDS) && (i-j>=0) ) {
        setPixel(i-j, red, green, blue);
      } 
    }
   
    showStrip();
    delay(SpeedDelay);
    i++;
    if(i=NUM_LEDS+NUM_LEDS-1) { i=0; }
  }
}

But even with that, I would have expected to see at least something happen. Does the strip work properly with other effects?

Any luck with the sketch? I haven't messed with my Arduino in a while, kind of forgot about it if I'm honest.  

Hope you enjoyed your holidays.

Happy New Year!

If I put in your code correctly (which I'm not sure I did), I have this:

#include <Adafruit_NeoPixel.h>
#define PIN 3
#define NUM_LEDS 81

Adafruit_NeoPixel strip = Adafruit_NeoPixel(NUM_LEDS, PIN, NEO_GRB + NEO_KHZ800);

void setup() {
  strip.begin();
  strip.show(); // Initialize all pixels to 'off'
}

void loop() {
  meteorRain(0xff,0xff,0xff,10, 64, true, 30);
}

void meteorRain(byte red, byte green, byte blue, byte meteorSize, byte meteorTrailDecay, boolean meteorRandomDecay, int SpeedDelay) {  
  setAll(0,0,0);
  int i = 0;
  
  while(true){
    
    // fade brightness all LEDs one step
    for(int j=0; j<NUM_LEDS; j++) {
      if( (!meteorRandomDecay) || (random(10)>5) ) {
        fadeToBlack(j, meteorTrailDecay );        
      }
    }
    
    // draw meteor
    for(int j = 0; j < meteorSize; j++) {
      if( ( i-j <NUM_LEDS) && (i-j>=0) ) {
        setPixel(i-j, red, green, blue);
      } 
    }
   
    showStrip();
    delay(SpeedDelay);

    i++;
    if(i=NUM_LEDS+NUM_LEDS) { i=0; }
  }
}

void fadeToBlack(int ledNo, byte fadeValue) {
 #ifdef ADAFRUIT_NEOPIXEL_H 
    // NeoPixel
    uint32_t oldColor;
    uint8_t r, g, b;
    int value;
    
    oldColor = strip.getPixelColor(ledNo);
    r = (oldColor & 0x00ff0000UL) >> 16;
    g = (oldColor & 0x0000ff00UL) >> 8;
    b = (oldColor & 0x000000ffUL);

    r=(r<=10)? 0 : (int) r-(r*fadeValue/256);
    g=(g<=10)? 0 : (int) g-(g*fadeValue/256);
    b=(b<=10)? 0 : (int) b-(b*fadeValue/256);
    
    strip.setPixelColor(ledNo, r,g,b);
 #endif
 #ifndef ADAFRUIT_NEOPIXEL_H
   // FastLED
   leds.fadeToBlackBy( fadeValue );
 #endif  
}

void showStrip() {
 #ifdef ADAFRUIT_NEOPIXEL_H 
   // NeoPixel
   strip.show();
 #endif
 #ifndef ADAFRUIT_NEOPIXEL_H
   // FastLED
   FastLED.show();
 #endif
}

void setPixel(int Pixel, byte red, byte green, byte blue) {
 #ifdef ADAFRUIT_NEOPIXEL_H 
   // NeoPixel
   strip.setPixelColor(Pixel, strip.Color(red, green, blue));
 #endif
 #ifndef ADAFRUIT_NEOPIXEL_H 
   // FastLED
   leds.r = red;
   leds.g = green;
   leds.b = blue;
 #endif
}

void setAll(byte red, byte green, byte blue) {
  for(int i = 0; i < NUM_LEDS; i++ ) {
    setPixel(i, red, green, blue); 
  }
  showStrip();
}

Unfortunately this just lights up the first LED on my strip white. It doesn't move, fade, etc. Perhaps I've put in the code wrong?

And no need to worry about a late reply, we've all got things to do.