两个用法都有，都是待在家里的意思， 只不过home的词性不同。
下面是找的权威例句，都是正规新闻，文章，词典中出现的
stay at home 中home 做名词
I would rather stay at home.
我宁愿待在家里。 from：《新英汉大词典》
They intend taking turns to stay at home and care for their three-year-old daughter.
他们想轮流待在家里照顾3岁的女儿。 from： CNN 文章
stay home中home 做副词
Her mother left her job with a local newspaper to stay home with her family.
她妈妈辞去当地报社的一份工作，选择留在家里。
from ：VOA: special.2009.10.18
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