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I don't think you need to expand it, but you can. Anyone who goes by two initials is usually called "P.J." or "PJ" in writing. Anecdotally, I knew a guy who went by G (for Gerard), and written down it was always G. That said, we weren't in the habit of transcribing our conversations.

If you don't want it to be just the letter G, I'd recommend writing it out as Gee, which is the (pretty) standard formal spelling of the name of the letter G. For more information on the names of letters in English, see:

Is there a formal spelling for the English letter names?

I'd leave it G. The letter G is pronounced the way you want it to be pronounced, so it's clear. If your test readers find it confusing (which I doubt they will), you could use some exposition the first time the character does that.

"Yeah, right, G," said Thomas, reducing Gary's name to a single letter, and not even the hard "G" of "Gary", but the weak, limp "G" of "gelatin" or "germ," as if Gary weren't worth the effort of the hard "G".

Okay, so that's rather overwritten, but you get the idea. :-)

The reader only needs to be told how "G" sounds once. You can put the explanation in-story, e.g. the character says or thinks 'I hate it when Bob calls me G, I can just hear him thinking "Gee whiz" whenever he does it' (this example is imperfect, you want to make the explanation completely unambiguous, but I think you get my point here). From then on, you can simply write G and everything is clear.

Depending on the style you're aiming for, just use the single letter.

For example, in Franz Kafka's novel The Trial (Der Prozess), the protagonist's name is Josef K. Throughout the novel, he is simply refereed to as K. - not just by the narrator, but also by other characters, who however mostly use the formal Mr. K. This is not unusual for Kafka - if I'm not mistaken, he does that in a few novels - but it should be noted that this gives a specific feel to the story and the relationship between reader and protagonist, which is perhaps not what you want.

Reading The Trial feels a bit like reading a report already, but the fact that character is just K. makes him more distant to the reader and emphasizes the report-like feeling. I don't want to spoil the end, but as is typical in Kafka's works (or at least the books I read), it doesn't end well for K. And although it's sad how it ends, I didn't really feel sad for the protagonist; his actual name isn't even known at the end.

Of course, this is subjective. Different readers get different things from stories and it's been ten years since I read it the last time. Because of that I recommend taking a look at The Trial.

Regarding sound and pronunciation

There is only so much you can do to give the reader the right idea how it is pronounced. In your example it is probably not even necessary since in English G is already pronounced exactly how you want it. However, if you want to be absolutely sure that the readers get how it is pronounced, because it is important to the story or it could get lost in translation*, you should simply mention it. Depending on the point of view and your style this could be done by the narrator

G didn't mind getting called just by his first initial, but he hated it when people mispronounced it just to mock him. Not a single pun he didn't already hear, and the get worse with time. Still G didn't grew tired of correcting people that it's pronounced just like the english letter, Gee.

or it could happen naturally in dialog i.e a character making a dismissive comment about G or a pun on G that only works if the pronunciation is wrong ( or right). Let the character correct them so the reader will know how it is pronounced. From that point on you can keep G in text, and the reader will still know its pronounced like the standard english letter G.

However this could be strange to read because it seems unnecessary to mention how G is pronounced. If it is not actually important to the story (because it has to rhyme or something) I would leave it to the reader.

*However if your story is not actually written in English then this might be of course important because even the same Latin letters are not pronounced the same in different languages. Take Kafkas K, for example. In German it would be pronounced Ka (with an A like in car) whereas in English it would be pronounced Kay (like in okay). I can't remember a single instance where this would make a difference in Kafkas novels but that doesn't necessary mean it's the same in your case.

It isn't unheard of to use single character names. Kafka's The Castle features a character named K. There is even a page on tvtropes describing single character names, which indicates the popularity of doing so: https://tvtropes.org/pmwiki/pmwiki.php/Main/OneLetterName .

'G' specifically is very common within certain subcultures, and is usually considered a sign of respect (short for 'Gangsta'). Is it your intention to subvert that cultural norm by turning 'G' into a diss (dismissal)?

You'd write it by spelling out the letter. In this case, that might be "Yo, Gee, you idiot!" or something similar. This is, not surprisingly, the same way you'd write someone reading a single letter or reading something like a serial code character by character (assuming the character doing so isn't trained to use a phonetic alphabet).

If you just write the letter G, most English-speaking people will pronounce it with a soft sound. I've never heard someone read the letter G as "guh" or whatever.

That said, does it matter? If a reader pronounced it in his head as "guh", would that hurt the story? If not, then I'd say just don't worry about it. If it matters, then make it explicit the first time you use it. Either just say, "He pronounced it with a soft G, like in 'gee whiz'." Or have a character say something about it.

I would write it G, but you could write it Gi or Gee, if you like either of those better. This may be a "matter of opinion" question.

In comments, there is a complaint "Gi" could be pronounced with a hard G, like "go" with a long "ee" instead of an "oh".

I agree, but the author can explain, the first time "Gi" is used, that it is pronounced like "Gee". (like "Joe" with "ee" instead of "oh").

The reason to use "Gi" and explain it, is to avoid confusion with the exclamation "Gee!" "Gi" IMO looks more like a name, and would be distinguished that way.

As for "Gigi" as a name, I have only heard it pronounced "Gee Gee".

A friend of mine is actually called Ge, but internationally she is known as 'G'. If you are not happy with just a single letter to indicate this character, you can go for a two letter 'name' that expresses the pronunciation you are happy with.

In case of my friend, it is the first syllable of her official first name, but around here (the Netherlands) it is not uncommon for a nick name to be the initial and a second letter which is not in the actual name but does support the pronunciation of the initial.

You could try changing the font and see how that feels, like as a cursive G, or try using bold, italic or other denotation. If one doesn't feel good about something that they write, that's a good indicator that it shouldn't be there, hence spelling it out would also be tabled if one isn't confident about it.

Does the character call him "G" or "Mr. G?"

I'm thinking of in Buffy the Vampire Slayer how Faith refers to everyone, so she calls Buffy "B".

And The Fonz in Happy Days calls everyone "Mrs. C." or whatever - it's like "as formal as he can do."

And do people use first-initials as nicknames if they don't rhyme with Bee? (I mean, 007 has Q and M, but they're designations more than nicknames: M for MI-5, Q for Quartermaster)

Also, some things due suit the eye more than the ear: Example = Wonder Woman. It makes sense on the comic page for people to call her "WW," but to say that aloud in a movie is just wrong - it's 6 syllables instead of 4, while the print version is 2 characters instead of 12.

So for your writing, do you "hear" the character called G by the other character? Do other names feel like they "sound right?" Then go with that.

But some writers are more visual or focused on the etymology in some word/name choices: the name Hermione Granger - many readers didn't know how it was pronounced (hence the scene in Goblet of Fire), but the meaning of it, Earthly Worker of the Granary (aka very mundane/muggleborn) might have overcome the "ear".

To be completely honest, I would also stick with the single letter. However, any awkwardness or confusion could be avoided by simply putting an introductory passage for this. "He spoke his name with just one syllable, 'G'. It was hard to get used to just speaking one letter, but it was definitely fitting."