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.