1. Rob saves a portion of every paycheck in the bank. He’s not saving for anything in particular, but one never knows when one might need some extra money all of a sudden. He’s saving for a rainy day
2. My mother always made me save some of my money in case I needed some unexpectedly. She made me save for a rainy day

Compare to: salt (something) away; on ice. The expression suggests that a rainy day is one when a person will not be able to earn any money, and so he or she needs to prepare for such a day by setting aside some money now.