The expression “to hang on by the eyelashes” means to be just barely able to retain one’s hold on something, literal or figurative; to be in a precarious condition.
Our ancestors, back in the seventeenth century “hung (something) by the eyelid,” rather than the eyelashes, indicating that they meant to keep that thing or subject in suspense.
Later, about a hundred years, “to hang by the eyelids” came into vogue with the meaning of our present expression.
One fighting to retain a spark of life, or one barely able to stave off financial disaster is, with equal impartiality, “hanging on by his eyelashes.”