Se lo llama el "double genitive". (A phrasal construction in English in which possession is indicated by the word "of" followed by the possessive form of a noun or pronoun, as in "a relative of mine" or "a friend of Pat's". Also called double possessive.)
Aunque se lo ve y se lo oye muy a menudo, es para evitar, máxime en la palabra escrita.
Mejor - "I met one of Jane's friends".
The structure "I met a friend of Jane's" is very common and natural, although in formal writing it would be less common. There might be situations in which one would use it conversationally rather than "I met one of Jane's friends," depending on the context and the emphasis one wished to give. For instance:
"He's a friend of Jane's."
In the above, "he's one of Jane's friends" doesn't quite sound right, although it wouldn't be "incorrect."