Not Barmby. It's someone obscure in the sense that they weren't/aren't exactly a big name, but not obscure enough that it will surprise you.
I was worried that you might have come straight out with it. Likewise with Teddy being one of the answers to question 2 (seeing as though one of his 3 hat-tricks was for you guys).
Here's a couple more questions. Question One :- Name 4 players who have played for 4 different teams that have won the European Cup though they have never won it themselves? Question Two :- Which player has played in the, Merseyside derby, North London derby, Milan derby, Munich derby and the North East derby?
Are you including the Champions League when you say European Cup? If so, then Ronaldo is one (Madrid, Barca, Inter & PSV). I don't believe he has ever won it.
Robbie Keane and Michael Owen have both played for 3 clubs that have won it, but not 4. So it's not them.