I think that you could go a variety of directions with this answer. On one hand, you could say that the title is significant because it quite succinctly tells readers what this story is going to be about. The opening few paragraphs confirm that the story is going to be about an old man. We are told repeatedly that the man is old. We are told that his skin is quite wrinkled and even shows those color changing patches that come with old age.
The old man was thin and gaunt with deep wrinkles in the back of his neck. The brown blotches of the benevolent skin cancer the sun brings from its reflection on the tropic sea were on his cheeks. The blotches ran well down the sides of his face and his hands had the deep-creased scars from handling heavy fish on the cords. But none of these scars were fresh. They were as old as erosions in a fishless desert.
We also find out early in the story that this particular old man is a fisherman that earns his living fishing on the sea. The title of the story is significant because it succinctly tells readers what to expect; however, I think the significance of the title can go deeper than that.
The title makes use of the word "and." The use of that word conveys to readers that there is a relationship between the old man and the sea. People in a couple are always referred to with "and." Jack and Diane for example. This could be why Santiago refers to the sea as a woman. He understands it as his partner in the way that a husband is partnered with a wife. As with most relationships, things can be good and bad. Santiago hasn't had a catch in over eighty days, yet he refuses to give up on the sea. He refuses to give up on his life partner, and the sea doesn't give up on him either. It's why he is granted the catch of his life; however, it doesn't come easy. Relationships don't come easy either, and despite losing the fish, Santiago will go to work with his partner again the next day.