Hi,
What do you mean, "I lost"?
I hit the ball in first! What do you mean I lost? (Quoted from a dictionary)
Why use quotation marks?
It's indirect speech, isn't it? True, in that scenario. But in this one, it's direct speech.
A: I had a winning position on the chess board, but I lost.
B: What do you mean, "I lost"?
A: My chess clock ran out of time before I could move.
Here's another approach, which is the thought I originally had. I admit that I may perhaps be on weak ground here, but here's what I was thinking. We sometimes use quotation marks to draw attention to the fact that we are sceptical about the word or phrase we are using. eg Tom told Mary that he 'loved' her. This suggests that the speaker doesn't believe Tom. That's why I put quotes around the phrase 'I lost', to show that the speaker was highly sceptical about the assertion. I should probably have put single quotes rather than double.
One final thought. If you don't put a comma after 'mean', the phrase 'I lost'' seems to be associated with 'what'. eg
A: What do you mean I lost?
B: I mean you lost your camera / faith / opportunity / etc.
Best wishes, Clive