What is the significance of Aaron's rod that budded?
To understand the significance, you first need to know that neither Moses nor Aaron had an easy time being accepted by the Israelites as their leaders. People had the mistaken notion that they made themselves leaders. After Korah failed to execute a rebellion against Moses, the people blamed Moses and Aaron for the deaths of Korah and his followers. "On the next day all the congregation of the children of Israel murmured against Moses and Aaron, saying, "You have killed the people of the LORD." Now it happened, when the congregation had gathered against Moses and Aaron, that they turned toward the tabernacle of meeting; and suddenly the cloud covered it, and the glory of the LORD appeared" (Numbers 16:41-42).
A plague broke out among the Israelites, which was only stopped by Aaron (who was probably in his 90's by this time) running through the camp to separate the plague victims from the healthy people.
To finally put an end to dispute as to who would lead Israel, God staged a demonstration. "Speak to the children of Israel, and get from them a rod from each father's house, all their leaders according to their fathers' houses-twelve rods. Write each man's name on his rod. And you shall write Aaron's name on the rod of Levi. For there shall be one rod for the head of each father's house. Then you shall place them in the tabernacle of meeting before the Testimony, where I meet with you. And it shall be that the rod of the man whom I choose will blossom; thus I will rid Myself of the complaints of the children of Israel, which they make against you" (Numbers 17:2-5). Aaron's rod not only leafed out the next day, it also had flower buds, flowers, and even ripe almonds on the same stick (Numbers 17:8). The man's name written on the sticks showed that it was the original one, but the growth in various stages of maturity also proves that no one had cut an almond branch and put it in its place.
Later Aaron's staff was placed in the ark of the Covenant as a memorial to the fact that God chooses whom He wishes to lead His people (Hebrews 9:4).