10분 더 자면 되겠네 = It'd be fine if you/I sleep ten more minutes => You/I might as well sleep ten more minutes.
* 10분 더 자면 = -면 is a verb form creating an if-like condition clause.
=> 10분 (10 minutes) 더(more) 자면 (if you sleep). The subject is omitted, so it's unclear who they are talking about.
* 되겠네 = 되다(work out; be all fine) + -겠(would, might) + -네(ending suffix expressing a mild surprise).
So the three combines to create a sense of "I see that you/I could sleep ten more minutes without a problem".
Another example:
- 옆집에서 호박을 몇 개 갖다 줬어. = Our neighbor brought us some pumpkins.
- 그래? 그걸로 호박죽 해 먹으면 되겠네. = Yeah? It'd be perfect if we make a pumpkin pie with it.