Les pronoms toniques sans verbe

Moi is the stress pronoun of the first person singular (je). In that single, one-word-sentence you can call it the subject, but that doesn't make much sense, since there is no verb nor anything else. It's just an abbreviated answer to the question in the form of an incomplete sentence.

Qui pourra le faire? - Moi. -- Who will be able to do it? - I. 

Here is the full version:

Qui pourra le faire? - Moi, je pourrai le faire. -- Who will be able to do it? - Me, I can do it.

Note that in English, sometimes the use of "me" mimicks the use of the stress pronoun in French. You'll often her "me" in the example above, which is, strictly speaking, incorrect. However, grammar is fluid and you can interpret that "me" as the stressed version of "I".

Moi/toi/lui/elle = Me/you/him/her - simple cases (French Stress Pronouns)

@Chris, English is not my native language althogh I know it better than French.

J'ai trouve un livre " Le Bon Usage. Grammaire Française. 14e édition " (Maurice Grevisse, André Goosse). Il y a un chapitre entier (page 844) "Les formes disjointes comme sujets". Et dans la page 847 on peut voir: " e) Le verbe manque ou il n’est pas à un mode personnel "

Certains me disent que je ne peux pas lire ou ne comprends pas ce qe je lis. Je ne suis pas sûr. Donc, je voudrais savoir les règles exactes (peut être, ce que des guides pédagogiques en disent).

I don't quite get what you are saying or what you are saying that the book is saying. Did my answer clarify the issue or not?

@Chris, first of all: thank you for your reply.

You write " you can call it the subject " and " you can interpret that "me" as the stressed version of "I" ". Of course, I can call and I can interpret. But It's not the same as " Yes, in that case "Moi" is the subject " or " No. It's not the subject, it's [some grammar term]. " E.g. a noun can be the subject, the complement etc. It's just an abbreviated answer, but it's still a sentence, and stress pronoun "Moi" must play some exact role (subject/complement/etc). Moreover, you also write " but that doesn't make much sense. " For me it basically looks like "yes, but no." And from that point of view your anwser is unclear and confusing. That is why I haven't given it a "like" so far. Sorry.

I interpreted the text in the above mentioned book as " Yes, this is the subject. " However, since my french level is quite low, I was not sure. But then I've got a confirmation from a credible source.

You can go nuts over what to call something and you'll learn exactly nothing. Parts of a sentence (like subject, verb, object, etc.) are only reasonably and uniquely defined in the context of a sentence. The single word "moi" doesn't constitute a sentence. So, there you go. Is it a subject? Yes, in the sense of the context implied by the previous sentence. No, because as a single word it can't be a subject.

For example: Je dois chercher qui ? -- Moi! (Who am I supposed to look for? -- Me!)

In this case, "moi" is the direct object in the context of the previous sentence. But, as a single, isolated word it is impossible to determine what grammatical function it has.

@Chris, ok, thank you. Then let me quote the above mentioned book (it can be found in the Internet Archive). On the page 844 one can see:

III. EMPLOI DES FORMES DISJOINTES

Les formes disjointes comme sujets.

following by the points a), b), c), ... h). And then on the page 847 one can see the point e):

e) Le verbe manque ou il n’est pas à un mode personnel :

Qui partira le premier ? MOI.

MOI is uppercased in the book. I interprete this in a way that MOI is the subject. Or in that case "comme sujets" is not the same as "sujets". And further "comme objets directs" is not the same as "objets directs" ? 

@Chris, you write " The single word "moi" doesn't constitute a sentence. " Isn't it called elliptical sentence (phrase, proposition elliptique)?