There are slight differences in meaning.
1) She was done cleaning her room. (A description of a state ... she was tall, she was sick, she was done, ...)
2) She has done cleaning her room. (incorrect grammar)
3) She finished cleaning her room. (A completed action in the past, during a time period viewed as closed)
... Last Saturday (closed time period), she finished cleaning her room, had lunch, and then went out for the day.
4) She has finished cleaning her room. (A completed action in the past during a time period viewed as open, viewed as extending until the present)
... Today (open time period), she has finished cleaning her room. She will join us for lunch and then she will go out for the day.
British and North American English vary slightly. Choice 4 in North American English will usually be "She finished."