Can a boy born in Shudra family become a Brahmin?

 Mahabharata talks about this.

Lord Shiva says

A shudra who is pure in his deeds, pure in his soul and has conquered his senses can obtain the status of becoming a brahmana. Brahma himself said this. It is my view that a shudra who is naturally pure in his deeds should be known as one who is purer than a brahmana. Birth, sacraments, learning and humility are not sufficient reasons for a person to be a brahmana. Conduct is the only reason. All the brahmanas in the world obtain that status because of conduct. O one with the beautiful hips! A person who possesses that good conduct obtains the status of a brahmana.

• 131, Dana Dharma parva, Mahabharata.

According to Yudhishtira any person who has the qualities of truthfulness, charity, lack of anger, lack of cruelty, nonviolence and compassion is a Brahmana and any person without these qualities are Shudras

The serpent asked, “Truthfulness and knowledge of the brahman can be found in all the four varnas. O Yudhishthira! Truthfulness, charity, lack of anger, lack of cruelty, nonviolence and compassion can also be found among shudras. O lord of men! You have said that the object of knowledge is beyond happiness and unhappiness. But there is nothing that is free from either. I do not think such a thing exists.”

Yudhishthira replied, “If these traits, not even found in a brahmana, are seen in a shudra, he is not a shudra. A brahmana in whom a brahmana’s traits are not found, is a shudra. O serpent! It is said that one in whom these traits are found is a brahmana. O serpent! One in whom these traits are not found, is marked as a shudra.

• 177, Ajagara parva, Mahabharata.

It's further explained in the text

Any restraint in anything then becomes unnecessary. O brahmana! One may be born as a shudra. But if he is established in his good qualities, he will become a vaishya, and even a kshatriya. Similarly, one who follows worthy conduct can become a brahmana.

• 203, Markandeya samasya parva, Mahabharata.

If Ved Vyas was Brahmin, then how was his son Vidur a Shudra?

 Firstly, I will tell about “Niyoga” and then I will come to the answer.

Niyoga is a sprocedure through which a widow can get a son by the help of a saint or Rishi and the child will be considered the son of her husband and the biological father will have no right on that. Niyog is restricted only for birth of child and not for fun. This is only for those women whoes husband was died without son. In the procedure, a saint or Rishi sees that woman from a distance and using his power of Niyoga, that woman get pregnant.

Now, coming to the story :-

Due to the untimely death of Vichitravirya, his queens Ambika and Ambalika were got the pregnancy with the Maharishi Ved Vyasa in Niyoga procedure. Ambika first mated with Vyasa through the niyoga process but kept her eyes closed during the process because his appearance and power frightened her. As a result, she gave birth to the blind Dhritarashtra. Later her sister Ambalika followed the same process with Vyasa, and knew to keep her eyes open. But she was pale with fear during the niyoga, and so gave birth to the albino (you can search about it on Google) Pandu. As both children have some abnormalities born by Ambika and Ambalika, so they sent their maiden Parishrami ( a Shudra ) in their place, who behaved appropriately during the niyoga process and gave birth to the exceptional Vidura.

Vidura’s father is Vyasa, a half-caste, and his mother a sudra maid. His status was always lower than his two half-brothers (Pandu and Dhritarashtra). That is why, Vidura is a Shudra.