Trivia: The bear that Jack Ryan brings home to his daughter is the exact same bear Bruce Willis brings to his son in Die Hard (both films were directed by John McTiernan). (02:06:00)
Visible crew/equipment: When the Red October sub has surfaced to let the crew get off because of the supposed nuclear leak, there is a brief shot where you can clearly see the camera and its filming crew in their tennis shoes on the deck. (01:45:52)
Continuity mistake: When Sally is saying that Stanley is keeping her awake, her grip on the bear's ear is changing between constant shots. (00:04:40)
Revealing mistake: Near the end, when Ryan and Ramius are talking, right before Ryan says "welcome to the New World", Ryan is facing Ramius and his thick hair should be blocking the hill in the distance behind him, but his hair is translucent and the hill is visible through his hair. (02:06:00)
Chosen answer: They use it to wipe the grease pencil markings off the screens.
We called it "the Duty Roll."