The only mention of any name in the story is this name, Jig. It is what the man of the relationship calls the woman of the relationship. He uses it conversation in the following exchange:
'It's really an awfully simple operation, Jig,' the man said. 'It's not really an operation at all.'
The girl looked at the ground the table legs rested on.
'I know you wouldn't mind it, Jig. It's really not anything. It's just to let the air in.'
The purpose of the character using the name here is to call her attention. The two have been discussing nonesense items such as the setting since the story began. Here, the man is obviously trying to restart a discussion that they were having before, and to continue a small argument or disagreement. Using her name is a way to keep her attention and reinforce the seriousness of the discussion.
Hemingway places the name here for a similar reason. It is unexpected, for the two have simply been referred to by their gender before this point. By inserting a name, Hemingway is getting the attention of his readers.
The name itself, Jig, would appear to be some sort of nickname, which implies to the readers that the two have been involved in a relationship for some time. By using a nickname, Hemingway is able to indicate this familiarity and still keep the identities of his two characters vague.
The jig is a dance in Ireland. Giving the girl the nickname of Jig may suggest that she is from Ireland and that she is, or has been, a lively, spirited girl, since the jig is an extremely lively dance. The girl may have performed this dance on more than one occasion during their travels. They have been together a long time. This is shown by all the hotel stickers on their luggage--possibly also indicated by the fact that they are carrying so much luggage. Jig is obviously an affectionate nickname. Hemingway is writing this story from an strictly objective perspective. That is why he doesn't give his characters names. The reader only knows what he can see, hear, and deduce from what he sees and hears. The nickname Jig may suggest the girl is Irish. She has to be from someplace, and evidently she is not from America. She speaks good English, so she would have to be frp, England, Scotland, Wales, Ireland, Canada, Australia or New Zealand--most likely from the European places since the American must have met her in European.
The name "Jig" is most likely used as a nickname of sorts - it illustrates familiarity between her and the man, but it also potentially has a deeper meaning. Judging by the way he brushes her comments aside and downplays the danger of the abortion, he could be treating her as a sort of joke, as it is used in the expression "the jig is up," where 'jig' means 'joke.'