The Bible tells us that she lived in Zarephath (Sarepta) which is located in Lebanon. She was not a Jewish woman. She was a Gentile. Jesus confirms this in Luke 4:25-26. Even so, she recognized Elijah as a prophet of the Living God when he requested some water of her. Water. That scarce, precious and necessary thing! Interestingly enough, she does not tell him that she does not have any. Instead she goes to get some and I have no doubt that the "little water" which she intended to bring was the last bit that she had.
As she goes to bring it, Elijah calls after her asking her to bring a bit of bread as well. The bit of water she had but the bread? Well... she didn't have that. Not yet at least. She would have some soon but a bit was all she would have. Was she being asked to give that up as well also?
She tells Elijah of her plight. It is not that she does not want to give. Her move to go get the water is indicative that she does. Also, instead of telling Elijah "I'm sorry, I don't have any." she is honest with him. Calling upon God as her witness, she states her condition. She and her son are at the end of the road. They are without hope. They will eat their last "meal" together and die.
And Elijah said unto her, Fear not; go and do as thou hast said: but make me thereof a little cake first, and bring it unto me, and after make for thee and for thy son. (emphasis mine)
For thus saith the LORD God of Israel, The barrel of meal shall not waste, neither shall the cruse of oil fail, until the day that the LORD sendeth rain upon the earth.
And she went and did according to the saying of Elijah: and she, and he, and her house, did eat many days.
And the barrel of meal wasted not, neither did the cruse of oil fail, according to the word of the LORD, which he spake by Elijah.
1 Kings 17:13-16
Now I ask you to consider this question very carefully. What do you think would have happened if the woman would not have heeded Elijah's words and she and her son would have consumed the bread first?
The question is not, can I afford to give. The question is, can I afford not to give.