According to some scholars, the expression comes from
Down Under. Cowboys spend a lot of their time outdoors
looking after animals. During cold winter nights when the
temperature dips, the men often cuddle up with their dogs to
keep warm. When it is slightly cold, they need the body heat
of only one animal to keep warm. Such a night is referred to
as `one dog night'. When it becomes extremely cold, the
men may need three dogs to keep warm. Such an extremely
cold night is referred to as `three dog night'. This practice of
embracing dogs for body warmth was quite common among
the Australian Aborigines.
Example:
Take extra blankets. According to the forecast, it's going to be a three dog night.
`Three Dog Night' is also the name of an American rock and
roll band. They were quite popular in the late 1960s and
early 1970s.