Mollie

From Memory Alpha, the free Star Trek reference.

You may also be looking for the name Molly.

Mollie was an Irish Setter, kept by Kathryn Janeway as a pet in the early 2370s.

Mollie, who was the runt of the litter, was rescued by Janeway from a dog pound on Taurus Ceti IV. Janeway chose her because she thought the dog "had spunk." (VOY: "Shattered")

Mark Johnson looked after Mollie when the USS Voyager was lost in the Delta Quadrant. (VOY: "Caretaker")

Janeway kept an image of Mollie and Johnson in her ready room. (VOY: "Eye of the Needle") Mollie had a litter of puppies shortly after Voyager's disappearance. By 2374, they had grown considerably and got along with Johnson's new kitten. (VOY: "Hunters" okudagram)