This umbrella is illegally used to hold Rubeus Hagrid's wand after he was expelled from Hogwarts School of Witchcraft and Wizardry in 1943. During his third year at Hogwarts, Hagrid was falsely accused by Tom Riddle to have released a creature on the school that killed a muggle-born witch named Myrtle. As a result of this, his wand was destroyed by the Ministry of Magic. Although it was destroyed, Dumbledore allowed him to secretly keep the shards of his wand in the shaft of a pink umbrella, the shards still possessing a good portion of the wand's original power. However, since he never graduated, Hagrid's magic was never flawless; so even though he was able to use the pink umbrella as a wand substitute, he didn't have the skill required for even simple feats of magic. This caused him to rarely perform the spells that he wanted to.
His wand was broken in half as part of his punishment when he was expelled. He wasn't meant to keep the fragments, so he uses the umbrella to hide them. Maybe the umbrella aligns the pieces and makes them more effective. Some fans also theorise that Dumbledore might have fixed Hagrid's old wand using the Elder Wand, because he knew Hagrid had been framed, and if that were the case Hagrid would have had to hide the fixed wand too. As to why he chose an umbrella to hide it in, no idea!
