Me and my friend, Tim, are gonna predict the winners of the next dancing with the stars!
I found this sentence from a book.
I wonder why we don’t use “I and my friend” since I think it must be the subject of “are gonna predict”
It can be a subject, but only in the sense that people can use incorrect grammar. It isn't correct grammar, for the reason you correctly state.
The correct usage is "My friend Tim and I are going to predict[...]". So you are correct, except common usage is to place the self last in the list, preceded by other pronouns, and putting any nouns first. So "My mother, Jane, he and I will get together tonight" is the typical order. (Note that in this sentence, my mother and Jane are two different people. If Jane were my mother, the sentence would read "My mother Jane, he and I will get together tonight." It's interesting how much a comma can change a meaning.)
On the other hand, it is the sort of thing that young people say to exasperate their elders. :)
No, me can't be a subject:
This is something no native speaker would say. It's out-and-out wrong. If you say this sentence, you sound like a caveman.
But the following sentence is very different:
Although this is perceived as informal and non-standard, lots of native speakers talk this way! It's a very natural sentence, and you'll hear people say this all the time.
So it's clearly a big mistake to say that these two usages are wrong for the same reason. Sentence 1 is ungrammatical—me can't be a subject. But me and my friend can! It may be considered informal and even non-standard, but sentence 2 is natural and grammatical.
Although sentence 2 is very natural, children are taught in school that it's incorrect. They're taught that they must say the following instead:
And so, as a student of Standard English, you should say it this way too. Instead of the natural order and case (me and X), you should use the opposite order and case (X and I). There's no reason like "me can't be a subject" or "I must appear last because it's more polite". The only reason is that students are taught to speak this way, and so you should, too.
Just keep in mind that you'll hear it the other way all the time, especially informally, and that when native speakers talk that way they're not speaking incorrectly.
(As an aside, the commas around Tim are unnecessary, but the difference between "me and my friend Tim" and "me and my friend, Tim" should be asked as a separate question.)
In this answer, the * symbol indicates that a sentence is ungrammatical.
The difference between "me" and "I" is well-addressed in this article. The sentence you have above is technically ungrammatical, because you wouldn't say:
However native English speakers would accept your original sentence as correct in colloquial English because in compound subjects even native speakers often fail to use the subject pronoun 'I'. This behavior is well explained in the article linked above:
As mentioned in another answer, however, the word order you used is slightly less common (though not incorrect), and more often you might hear either of these sentences:
This is interesting - in speech, you will often hear people doing this, and it sounds natural - to me, as a native speaker, there's nothing wrong with it, even though it uses "me", which you normally don't find in the subject.
To answer your question:
It seems to me that
This might be a way for the studious person speaking to distance himself from his dissolute friends who are off to the pub.
I believe that the above is both idiomatically and grammatically correct. Question is - is "me" considered part of the subject? If it is not, can someone tell me what it is?
On a related note - nobody would say "I and my friend are going to predict...". So let me ask the question differently. Clearly we can say:
Then ought there not to be a syntactically correct way of changing the order? My first choice would be:
but I accept that this doesn't sound quite right. However, I'm pretty sure that
is correct - and follows the same approach as the first example above. In other words - "Me" used at the start of the sentence for emphasis, but repeated in another word which becomes the "actual subject" (in this case, "we"). It is clearly better than
Of course in the quoted sentence, this is only one of the problems.
The use of the comma: "Me and my friend, Tim, " implies one of two things. Either the speaker has only one friend, and that friend's name is 'Tim'; or the speaker is addressing someone called "Tim" while explaining that he and his friend are going to do something.
Then there is "gonna"...
And finally "the winners of the next dancing with the stars", where Dancing with the stars, which is the title of a TV show, needs to be shown to be a title by either using italics, or putting the name in quotes; either way, as a name it ought to be capitalized (and quite probably fully capitalized: Dancing With The Stars is how it is shown on the owners' website).