On first consideration, hanging on can seem only a slight improvement on giving up. Or it is the final stage before you admit defeat. A person is hanging on by their fingernails or hanging on by the skin of their teeth.
It can be tempting to think of defeat as inevitable.
I prefer to think that instead of hanging on, the protagonist, character, or next door neighbour is taking a brief pause in their fight against a disease, a difficult financial situation, or a family crisis. The pause is to gather your resources, gird your loins (as the old saying goes). On a side note, this expression comes from the days when men and women wore long flowing robes on a daily basis. In preparation for battle, or fleeing an invasion, the robes had to be tucked up to facilitate speedy movement either towards or away from the action. Hanging on is the modern equivalent of girding your loins.
Or perhaps be inspired by Winston Churchill's closing remarks in his
well-known commencement speech: