1 (Gertrude)  
2 (Gigi)  
3 (Greedy)  
4 (Greta)  
5 (Gata)  
6 (George)  
7 (Gwyn)  
8 (Gaea)  
9 (Gardyloo)  
10 (Gail)  
11 (Graff)  
12 (Giselle)  
13 (Gaige)  
14 (Gus)  
15 (Gunter)  
16 (Gemina)  
17 (Galadriel)  
18 - in the comments (Garnet)

@when she felt her wings unfold  
I’ve always liked to imagine, at the risk of ripping off Tolkien somewhat, that most names in a theoretical “Proto-Griffish” started with voiceless labiovelar /kʷ/ (calling to mind onomatopoeiae like ‘quack’ or ‘squawk’, as primitive griffons likely couldn’t form more complex words) that over time became prenasalized /ᵑkʷ/ early into Proto-Griffish (as a distinction from voiced labiovelar /gʷ/, which would develop in a different way), and which in turn eventually caused them to lenite finally into voiced stop /g/; owing to their having beaks, labial consonants might not have come easy to the griffons until much, much later eras following extended contact with other races such as ponies.
 
For a rough example of this development: Guto < *Nkutō < *Ŋkwəttau < *ᵑkʷak-tʰau

Also, to me the “G rule” is partly because griffons can’t pronounce plosives or fricatives with their lips like ponies can. What with the beak and all.
 
However, they have a very flexible syrinx and glottis setup to compensate. In my hedcannon, griffons have whole orchestra of deep windpipe syllables with lyrical, trilling and piping overtones that more conventionally mammalian creatures find impossible and try to approximate with a voiced velar stop, /ɡ/.
 
So in other words their “G”s all sound subtly and meaningfully different, at least to other griffons.
 
(Alt: their more complex historical culture probably had an awful lot of specialized rank-and-status honorific prefixes, but these days everygriffon calls everygriffon else by the “low-status jerk griffon” one. So maybe there’s not an awful lot of variation any more ☹)

I’m partial to Gardyloo for 9, thanks to his waste-water defenestration habits…  

 
However, MLP.wikia prefer ‘Godot’. Worth the wait?

The names below are either official or generally recognized names.
 
#3 – Gimme Moore (I don’t know if I should hate it or love it.)  
#4 – Greta  
#12 – Irma (cards call her Giselle)  
#16 – Natalya
 
Anyone got any others?

#2 Gigi  
#6 George  
#7 Snowy  
#1 Gertrudi  
#10 Gladys  
#15 Mr. Grumpy  
#5 Galena or Guienevierre

#10 Silver Quill’s Relative?

@RyuSpike
 
“Grace” or “Gracie” as a sarcastic name, because she ran right into Greta.
 
“Gillian” because it’s a nice name.

@Stotter  
Hehe best new reaction image of season 5.

@Blissey1  
I know, it just looked like a good excuse to use that image.

@Stotter  
well yes, I clearly said “imo”

@Blissey1  

@RyuSpike  
How about “Guts”?

@Stotter  
Hehe, all of the Griffon names have to start with a G…

@Stotter  
the g rule is fucking stupid, imo.

@RyuSpike  
Forgot the G rule.

@RyuSpike  
I’m fine with Tigerbutt.

#5 is the delegate of Griffonstone in “Princess Spike”. We need to give her a name now. She has stripes so that makes her pretty unique. And no. Naming her “Stripes” or “Tigerbutt” should not be an option.
 
But if we do end up calling her those things, remember who came up with it.

Update for #1 and 2:
 

@Stotter  
I’ve been saying this for ages whenever people’s gryphon OCs are inevitably a tan feline half and white eagle front half, and its super satisfying to have a huge dose of proof positive with this.

It’s neat how the bird portion of griffons isn’t just the traditional eagle. Based on #5, the feline portion might have variety too.

@Background Pony #7ACC  
I believe they`re playing street dice, not any kind of DnD related thing

@Background Pony #7ACC  
Or: #1: Guy, #2: Gax

@SunShowerIsMyMare03  
Please tell me you’re just an alias of RainbowDash2014 and not ANOTHER person doing the Kanaya Typing Quirk Thing or I think I might have to shoot myself.

@Ping_chan  
Have you seen the link posted in Cheezedoodle’s comment? There’s plenty of them. It’s hardly limiting.