And there you GO.
Answer: Arsenius Jigger
[>>Sahil got that one. =D]
The smarter ones might have guessed this by checking up book-lists that Hogwarts students receive every term. What makes it easy is that throughout the book, only one Potions book had been recommended. And that's th one in Book 1.
However, if you want the links between the answer and the poem, here you are:
The ‘she’ is Perenelle Flamel, wife of Nicolas Flamel. The ‘he’ is Nicolas himself. Bumblebee translates to Dumbledore in Latin. He finished off Grindlewald in 1945. Throughout all the books, the only other pieces of information regarding Perenelle and Grindlewald are that she lived to be 658 (the poem hints at her age) and that Grindlewald was killed in 1945 (the poem hints at his death).
If you play around with both these numerical values, you’ll find that,
6 + 5 + 8 = 19
1 + 9 = 10
1 + 0 = 1
and,
1 + 9 + 4 + 5 = 19
1 + 9 = 10
1 + 0 = 1
If you can actually get to this part, you’ll see, that in a numerical sequence, ‘A’ is the first letter and ‘J’ is the tenth letter.