My mother has told me that bichon frises were taken on ships by Italian, Spanish and French sailors. Wikipedia agrees with this but doesn’t really give an explanation why.
Animals have served a wide range of duties in shipboard life. Cats and Dogs would be used as Ratters, ie. killing any rats that got on board. Dogs could be used to hunt depending on the breed and many provided companionship. Some have even been awarded medals for their service from the PDSA (People's Dispensary for Sick Animals) There's quite a lot of interesting stories on the animals of various wars that they have awarded medals to. Simon the Cat and Sinbad the dog are a couple of sailing animals who received medals. Dickin's Medal List
As for the Bichon breed itself it was generally associated with status, so captains and nobility would bring them along on trips to represent how important they were. The are also very intelligent and are able to be trained for a multitude of tasks, so they could just as easily be used to help keep the ship free of infestations.