"player" was given as a possible translation for μέλος, but "our new player" wasn't accepted. Is it wrong?
Not really, since there is no context issue. I just added it, thank you ^.^
But what is a recent member? Does it mean a member who has recently joined our club etc.?
I would think that's exactly what it means!
Can you change the order and say 'το πρόσφατο μας μέλος'?
Yes you can but I think an extra accent is necessary on the last syllable of "πρόσφατο": "Το πρόσφατό μας μέλος".
I think it's right.
On further thought, in English, it could refer to a member who has recently died. i.e. he was a member until recently. I just wondered what it would mean in Greek.
Would "The new member" be acceptable? "The recent member" doesn't have the "recent entry" meaning in English
That's correct as well.