What is a good thesis statement for D.H. Lawrence's "The Rocking-Horse Winner"?

There are several key themes running through the story by D.H. Lawrence called "The Rocking-Horse Winner," and they could all be written about effectively in an essay, which is what I assume you intend to do with your thesis. 

One possible theme concerns materialism, the relentless need Hester has for more money with which she buys more things. From the beginning we see that she is miserable because she chooses to be. Though she has "all the advantages," she is not content. She is not content with her husband, her children, her house, or her things. What she thinks will make her content is money; however, every time she gets money she immediately spends it and wants more. Even the house senses her need for more money:

And so the house came to be haunted by the unspoken phrase: There must be more money! There must be more money! The children could hear it at Christmas, when the expensive and splendid toys filled the nursery. Behind the shining modern rocking-horse, behind the smart doll's-house, a voice would start whispering: "There must be more money! There must be more money! "

Her son,  Paul, tries to fill that need, and finds a way to get more money for his mother, something neither of his parents (who have expensive tastes and consistently spend more than they earn) has been able to do. After he wins it and manages to get it to her, Hester is not appreciative; she wants more and she wants it now.

In the end, she loses something that should have mattered to her very much, her son, but she cannot even grieve for him because she is too consumed with spending the money he left his family. We are sadly confident that spending that money will not bring her any more satisfaction than anything else has. So, a possible thesis might be that materialism, the hunger for money and things, is a need that can never be satisfied. 

Here is another idea. The first line of the story makes it clear that Hester is

a woman who was beautiful, who started with all the advantages, yet she had no luck.

This idea of "luck" is also connected to money by, of course, Hester. Though she started with everything, she constantly whines that she has no luck. When Paul misunderstands and thinks luck is lucre, the connection between luck and money is clear. A possible thesis statement might be that Hester's lack of luck is actually her lack of money. Another version of that is something like this: Having luck and having money are inter-connected in this story.

Here are a few other possible themes to think about:

• Just as there is "never enough money" in this story, there is also never enough love or enough sacrifice to satisfy Hester.
• Paul strives for money to win his mother's love and approval; she consumes the money but never gives her love and approval.
• The parenting roles in this story have been reversed, as demonstrated by Paul's frenzied attempts to provide for his family.
• Happiness (or contentment) cannot be found in luck or in money.
• Being too generous (Paul) is as costly as being too greedy (Hester); both Paul and Hester lose their lives, one literally and one figuratively. 

Whatever idea you choose, whether it is one of these or one of your own, your thesis should reflect the points you want to make. Choose specific examples and details from the story to support your points, and you will have a logical  outline for writing an effective, well organized, and well supported essay.

What could be a thesis statement for "The Rocking-Horse Winner"?

I think it is possible that this question is mixing up two ideas. The question asks if there is a specific "one" that is important to know. The wording of the question makes it seem like it is asking if there is an important thesis statement regarding this story. I believe that the second half of the question is asking about themes in this story. Yes, there are several themes present in this story that are important to know. A thesis statement can be written about this story and those themes. Thematically, the story contains a deep criticism of materialism and greed. There is also a theme of family. Your job as a thesis writer is to make an argument to your reader about those themes. It's entirely up to you what you want the thesis argument to be. I would recommend a thesis that steers readers toward seeing how Paul's efforts actually made things worse for his family: "Although Paul believes that he is helping his mother out by giving her what she wants, he is actually encouraging her materialistic and greedy tendencies."

How can the introduction to "The Rocking-Horse Winner" be summarized?

“The Rocking-Horse” winner by D.H. Lawrence introduces a woman he does not name. He only briefly mentions that she was both beautiful and had advantages, but the main important characterizing feature at first appears to be her lack of luck.

Lawrence goes on to describe this feature in her marriage and motherhood. In her relationship with her children, it becomes clear that her lack of luck is in fact a basic lack of love. Perhaps the husband reacted to this lack of love in the same way the children do.

What is interesting is that the woman does not succumb to the hardness in her heart, which she seems to have no control over. Instead, she tries to override it by overt expressions of affection.