Different airlines have different rules, but if you are looking at extending more than 24 hours you are pretty much out of luck. Some will allow for same day change as a standby for no charge or a small charge. Others will do the same thing as long as the flights are within 24 hours of each other.

Assuming you purchased one of their cheapest (ie restricted) fares, they most likely would stick to their guns and make you pay for a change, unless you spoke to someone at the check-in counter who was willing to make an exception, which would be highly unlikely in my opinion.