Although you asked about me vs m’a, you also changed the verb tense between the two options you offered. So I have to assume you know more than you’re letting on.

As I’m guessing you know, m’a is the elision of me + a, the a being the third person singular conjugation of avoir to form the past participle. Elle m’a acheté le déjeuner = She bought me lunch.

The second option is in the imperfect past tense, which is not a compound tense: avoir is not used here, but the elision doesn’t go away. me/te/se always elide with a verb that begins with a vowel. So it actually would be Elle m’achetait le déjeuner, which can translated myriad ways such as: She would buy me lunch, she used to buy me lunch, she was buying me lunch, etc. Imparfait can be nuanced.

But you’re trying to describe a simple action that happened once and is now completed, so you want the past participle (participe passé), the first option: Elle m’a acheté le déjeuner.

The main problem with your question are the suggestions you provide.

The correct tense would be the passé composé used in the second suggestion.

The sentence “elle m’a acheté le déjeuner” is correct, you will be understood, however if feels like you are implying the girl has bought you something to eat for lunch (like a take-away).

While they are both literal translations of “she bought me lunch” (although in two tenses but understandable), a more accurate translation should be “elle m’a invité(e, if you are a girl) à déjeuner”.

It is indeed more frequent in French to say you have been invited to lunch rather than have had one bought. However, be careful for there is a little double entendre with this translation : one can either say it before going to lunch (which means she literally invited you and you do not know whether to go), or after (which then means you went to lunch with her and she paid for both).

As for the m’a/me issue, the m’a is a contraction of me + a (avoir, present 3rd person singular). “Elle me a invité” cannot be said, you have to contract. This is also true for any other verb which starts with a vowel (“je m’effondre par terre”) and every pronoun (“elle s’insurge contre la reine”).

I hope I could help you.