In the lecture 'Lessons from Seerah in the light of Surah Alaq' an incorrect reason was given for the grandfather of Rasulullah SallalahuAlaihi wa Sallam vowing to slaughter one of his sons. The correct reason and version of events is as follows:

For many years, the people of Quraysh had to fetch their water from far away. One day 'Abd al-Muttalib was very tired from doing this and fell asleep next to the Ka'bah. He had a dream in which he was told to dig up Zamzam. When he woke up he was puzzled because he did not know what Zamzam was, the well having disappeared many years before he was born. The next day he had the same dream, but this time he was told where to find the well.

 

'Abd al-Muttalib had one son at that time, and together they began to dig. The work was so difficult that Abd al-Muttalib made an oath to Allah that if one day he were to have ten sons to help him and stand by him, in return he would sacrifice one of them in Allah's honor.

After working for three days they finally found the well of Zamzam. Pilgrims have been drinking from it ever since. The years passed by and 'Abd al-Muttalib did have ten sons. They grew into fine, strong men and the time came for him to keep his promise to Allah. He told his sons about the promise and they agreed that he had to sacrifice one of them To see which one it would be, they decided to draw lots, which was the custom of Quraysh when deciding important matters. 'Abd al-Muttalib told each son to get an arrow and write his own name upon it and then to bring it to him. This they did, after which he took them to the Ka'bah where there was a man whose special task it was to cast arrows and pick one from among them. This man solemnly proceeded to do this. On the arrow he chose was written the name of 'Abd Allah, the favorite son of 'Abd al-Muttalib. Even so, the father took his son near the Ka'bah and prepared to sacrifice him.

Many of the Quraysh leaders were present and they became very angry because 'Abd Allah was very young and much loved by everyone. They tried to think of a way to save his life. Someone suggested that the advice of a wise old woman who lived in Yathrib should be sought, and so 'Abd al-Muttalib took his son and went to see if she could decide what to do. Some of the Meccans went with them and when they got there the woman asked, 'What is the price of a man's life?'  They told her, 'Ten camels', for at that time if one man killed another, his family would have to give ten camels to the dead man's family in order to keep the peace among them. So the woman told them to go back to the Ka'bah and draw lots between 'Abd Allah and ten camels. If the camels were chosen, they were to be killed and the meat given to the poor. If 'Abd Allah was picked, then ten more camels were to be added and the lots drawn again and again until they finally fell on the camels.

Abd al-Muttalib returned to the Ka'bah with his son and the people of Mecca. There they started to draw lots between Abd Allah and the camels, starting with ten camels. Abd al-Muttalib prayed to Allah to spare his son and everyone waited in silence for the result. The choice fell on Abd Allah, so his father added ten more camels. Again the choice fell on Abd Allah, so they did the same thing again and again, adding ten camels each time. Finally they reached one hundred camels, and only then did the lot fall on the camels. Abd Allah was saved and everyone was very happy. 'Abd al-Muttalib however, wanted to make sure that this was the true result so he repeated the draw three times and each time it fell on the camels. He then gave thanks to Allah that He had spared Abd Allah's life. The camels were sacrificed and there was enough food for the entire city, even the animals and birds.

Reference: 
Lessons of Seerah Part 6 - In the Light of Surah Alaq