'a friend of Mike' or 'a friend of Mike's'

number 1 is correct... :-)

Both are correct, but the second is more idiomatic, as you already know.

If this is a repeat question, my bad. I'm on my fifth rye and ginger. I'll have two for you, Aspara Gus.

victoIf this is a repeat question, my bad.

Oh, no, I said as you already know because you’re a native speaker.

victoI'm on my fifth rye and ginger. I'll have two for you, Aspara Gus.

I’d better not tonight. Next time!

in sentence of number2 --- he is friend of mike's …. "friend of" is a possession telling that is mike's friend then why do we need use another possession mike's in the end of sentence?!....

aicahcutewhy do we need to use another possessive at the end of sentence?

It’s not grammatically necessary; it’s just idiomatic. The double possessive should probably be avoided in formal writing, however.

So in formal writing, you'd use this?

He is a friend of Mike.

victoSo in formal writing, you'd use this?
He is a friend of Mike.

I would aim for something a bit more natural, like He is Mike’s friend or He is one of Mike’s friends.

But without a recast, which is more acceptable in formal writing—a friend of Mike or Mike's?

I’d have to say the former.