The sentence should be "My aunt invited my friend and me to dinner."
The first person singular pronoun "I" is used when it is the subject of a verb. The first person singular pronoun "me" is used when it is the object of a verb or preposition. In the sentence above, "my friend and me" is the object of the verb "invited." ("My aunt" is the subject.)
When using a compound subject or object (one with two or more people/objects like "my friend and I/me"), sometimes it can be difficult to determine which personal pronoun to use. One thing you can try is using only the pronoun. For example, in "My aunt invited my friend and I to dinner" try removing "my friend and." This leaves you with "My aunt invited I to dinner" which may sound more obviously wrong.
Here are more examples of 'me' and 'I' in sentences.
'Me' in object position:
'I' in subject position:
I hope this helps.