Out of the blue is an idiom to describe something or someone as coming out of nowhere, or with no notice or warning. Another version of the phrase is out of the clear blue sky.
Out of the woodwork is an idiom when something or someone appears unexpectedly, but usually as a result of an action. When someone has won the lottery, then friends and long-lost family members might come out of the woodwork.
Most attribute the phrase to bugs that can survive unseen in the floorboards or walls of a house.
In short, out of the blue is unexpected with no cause, while out of the woodwork on the other hand is unexpected with an attributable cause.
Joking aside, the man who got Northland’s emergency helicopter service up and running, and has been involved with many community initiatives and sports developments, and has been a local government representative for many years, said being made a Member of the New Zealand Order of Merit came out of the blue. [The Northern Advocate]
“We took the official visit at Pitt and everything looked good, then out of the clear blue sky he shows up at my office early [Tuesday] morning.” [Pittsburgh Post Gazette]
Since late last week politicians of all persuasions have sprung out of the woodwork and said it is “essential for a democracy” for TDs to be allowed to say things under privilege and for them to be reported by the media. [The Irish Times]