Without more context, it's hard to give a definitive answer. However, here's one meaning that no one else has mentioned. In visual art, a "study" is a series of works that explores and investigates the nature and use of a particular visual component. Thus, an artist might embark on a "study of motion" or a "study of light". In this context, a "blue study" would be a study of the color blue and the artist might experiment with different shades of blue in a series of pictures (paintings, drawings, photographs, whatever). Usually, the "study" is in a single medium (oils, acrylics, watercolor, pen and ink, photographs) but it could span across multiple media if the artist is so inclined. However, the phrase "a blue study" is perhaps less commonly used than "a study in blue".

I've never heard this expression. I've seen the expression "a brown study" which simply means "deeply absorbed in," but I've only seen it in books. It's rather old-fashioned and literary. In "The Resident Patient," from "The Memoirs of Sherlock Holmes," Watson says: "Finding that Holmes was too absorbed for conversation, I had tossed aside the barren paper, and leaning back in my chair, I fell into a brown study."

It's not an idiom. It may refer to "study" as a room in the house where people often work and have a desk. So if the study was painted blue, it could be a "blue study". "Blue" is used to mean sad or depressed and also, in the context of art, sexually explicit. So a "blue study" could be an artistic study of something sexually explicit. I'm just guessing though.

Just a wild guess: There is a flashcard app that some people use for studying called, "StudyBlue."

Unfortunately I don't .. recollection, reminding or something like this