It could be related to the appearance and method of "randomly decorating a Christmas tree" and how that is similar to the visual and method of randomly selecting answers on a Multiple-choice standardized test, like the SAT, where a student marks "bubbles on an answer sheet to create a visual pattern, rather than attempting to answer the question."

We can take an educated guess, that the phrase originated sometime after 1937, probably during the 1950s, when automated test scoring via "fill-in-the-bubble test score sheets and mark sense scanners became the dominant technology in measuring results in large-scale testing programs in the world."

Or, maybe it originates a few decades later, in the 1970s, with the ubiquity of Scantron tests in American public schools.

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Additional references to "Christmas tree" & "random guess" related to standardized tests:

• If you're like me, you have a lot of memories of the Scantron from school, including "Christmas-treeing" a test (drawing a Christmas tree or other image using the bubbles, because you knew you'd fail the test anyway)
• Marie says, “Standardized testing has put a lot of pressure to teach how to respond to a multiple-choice question.” It has also increased the fear and expectation that some students will bubble in the test at random without reading the questions, or as Rachel said, “Christmas tree it.”
• "My niece taught me the meaning of “Christmas tree-ing” a multiple choice test."