In Flannery O'Connor's The Life You Save May Be Your Own, why does the elder Lucynell allow Mr. Shiftlet to live with her and the younger Lucynell?...

When Mr. Tom T. Shiftlet, as he calls himself, shows up at the farmhouse of Lucynell Crater, Lucynell first sizes him up to determine whether or not he's a threat. Based on his physical appearance, she determines Mr. Shiftlet is a drifter and, therefore, not a threat to her or to her daughter, also named Lucynell. Based on their conversation, Lucynell decides Mr. Shiftlet could be of use to her and the younger Lucynell in several ways. She allows him to move in with them in the hopes that he will marry the younger Lucynell. 

Once they've been talking for a few minutes, Lucynell asks Mr. Shiftlet what carries in his box. He claims the box holds tools. Lucynell says he can work for her and that she will give him food and lodging, but she cannot pay him. This continues the conversation, and Lucynell learns more about Mr. Shiftlet's opinions on humanity.

The first indication readers have that Lucynell is considering Mr. Shiftlet as a potential husband for her daughter comes after young Lucynell trips and falls to the floor and Mr. Shiftlet picks her up.

"Is she your baby girl?" he asked.
"My only," the old woman said, "and she's the sweetest girl in the world. I wouldn't give her up for nothing on earth. She's smart too. She can sweep the floor, cook, wash, feed the chickens, and hoe. I wouldn't give her up for a casket of jewels."
"No," he said kindly, "don't ever let any man take her away from you."
"Any man come after her," the old woman said, " 'll have to stay around the place."

Mr. Shiftlet seems to sense an invitation, or at least an opening here. 

"Lady," he said, jerking his short arm up as if he could point with it to her house and yard and pump, "there ain't a broken thing on this plantation that I couldn't fix for you, one‑arm jackleg or not. I'm a man," he said with a sullen dignity, "even if I ain't a whole one. I got," he said, tapping his knuckles on the floor to emphasize the immensity of what he was going to say, "a moral intelligence!" 

The two characters appear to be acting in a "double con," neither knowing the other is attempting to con them. Mr. Shiftlet teaches young Lucynell a few words, and he begins to fix the elder Lucynell's car, essentially demonstrating he is indispensable to the farm. 

A few days later, the elder Lucynell lays her intentions out on the table:

That night, rocking on the porch, the old woman began her business at once. "You want you an innocent woman, don't you?" she asked sympathetically. "You don't want none of this trash."
"No'm, I don't," Mr. Shiftlet said.
"One that can't talk," she continued, "can't sass you back or use foul language. That's the kind for you to have. Right there," and she pointed to Lucynell sitting cross‑legged in her chair, holding both feet in her hands.
"That's right," he admitted. "She wouldn't give me any trouble."
"Saturday," the old woman said, "you and her and me can drive into town and get married."

Though readers can see from the first time the elder Lucynell inquires about Mr. Shiftlet's skills and interests that she might be looking for a way to marry off her daughter, it's also clear by this point in the story that the elder Lucynell has no intention of sending the younger Lucynell off with Mr. Shiftlet (or anyone else). She wants Lucynell to marry someone who will stay on the farm, and who will need them as much as they need him.

The two do marry, but the marriage is, of course, doomed. Mr. Shiftlet's con was to get the car and as much money as he could from the elder Lucynell. Getting what he could, he drove away in the car, leaving the younger Lucynell, his bride, asleep in a cafe.