First, whether a G is pronounced as hard or soft seems fairly random, so it’s normal to think it’s pronounced like “Gift” without the T or like “Jiff” the peanut butter. Some people have been arguing that g-words are always pronounced with a hard G, and that’s not true…giraffe, Germany, digit, there are hundreds of Gs pronounced with “J” sounds. You will notice, in fact, that the letter G is pronounced with a J sound.
What’s really silly is to have a big argument about it…in the song, I was making fun of the big argument, not trying to become a part of it.
However, I generally find that, when you want to pick between multiple accepted pronunciations, it’s good to go to the people who created the word and ask them. I mean, they made it up, they should have say. The people who created the GIF format say “jiff,” so that’s what I say. Just like I say “Ah-see-oh” because that’s how JK Rowling says Accio, and “Wih-kee-pedia” because that’s how Jimmy Wales says wikipedia.
BUT!
In the end, both pronunciations are accepted, and it is common in language, for words to have multiple accepted pronunciations.
So, basically, say whatever you want.