She got her son ------- (sleep ) on the floor last night

to sleep是对的!
1)get+N.+adj.表示"使...成为"
e.g. I want to get this matter clear.
2)get+N.+done.表示"使..."
e.g. She got her hair cut.
3)get+N.+to do.表示"使...做"（有说服的含义）
e.g. I'll never get my mother to agree.
4)get+doing.表示"开始"．
e.g. Get moving,please!
此句显然是说服对方做某件事情．故用to sleep.
她昨晚让他儿子睡在了地板上．