Cyrus Ramsey was a fictional Human whose existence was part of a hallucination Ensign Hoshi Sato had when she used the transporter for the first time in 2152.

While in the pattern buffer, she experienced several hours of time on the Enterprise NX-01, during which she talked about her fear of the transporter with Tucker, Reed, and Mayweather in the mess hall. It was Tucker who related the details of Ramsey' story:

"Madison, Wisconsin, May, I think, 2146. He was a test subject for the first long range transport: just 100 meters. Something went wrong with the pattern buffer – he never re-materialized."

The three of them explained that Ramsey's disappearance became legend and he was the subject of many ghost stories in which he would re-materialize in front of someone. Even Dr. Phlox, who had only been stationed on Earth for nine months, was aware of Cyrus Ramsey. Upon hearing the story, Sato feared that her molecules were destabilizing and that she would disappear, just like Cyrus Ramsey had.

In reality, Sato's molecular pattern was trapped within the transporter pattern buffer for 8.3 seconds. The transporter was affected by storms on the surface of the planet she had been on with Tucker, so Reed had trouble re-integrating her matter stream. After finally emerging from the transporter, Sato realized that the entire experience had been a hallucination. She mentioned her concern that she would be "the next Cyrus Ramsey". Tucker and Reed, however, did not recognize the name. (ENT: "Vanishing Point")

This character was only mentioned in dialogue.
USS Enterprise Owners' Workshop Manual states that Cyrus Ramsay [sic] was one of the first test subjects for long-range transports, who failed to materialize after experimental transport. The accuracy of this is at best dubious, however, as within the same section, it describes Hoshi Sato's experience out of phase in "Vanishing Point" as if it were real, rather than merely a delusion she experienced while being trapped in the pattern buffer.
The Star Trek Encyclopedia (4th ed., vol. 2, p. 203) described this character as a nonexistent person.