A dream animal is always one you should hunt, but one you should be particularly careful to hunt "properly."
What I mean by that is that you should make sure that the hunt is conducted in such a way that you will remember the entire experience with pride. I can think of no better way to destroy a "dream" than to approach it the wrong way. For example, say your dream animal is a cape buffalo, and you go and hunt one in a small enclosure from the back of a truck. My guess is that your dream will die a minute or two after you pull the trigger. But chase buffalo around a large area, track them on foot over a period of days, get busted a few times, and finally get up close and personal and seal the deal - that's a dream animal that will live in those dreams forever.
You mentioned giraffe. I've shot a number (I seem to enjoy it, for reasons which escape me and most certainly escape my wife). Some hunts have been so easy that they weren't any fun at all (other than maybe as a marksmanship exercise), while others have been challenging and interesting. You can't always determine how the hunt will go - sometimes a giraffe comes easy, other times it comes hard, but whatever the case, make sure you've done it properly. On foot. In a large area. Sneaking up on the animal. Trust me, it's not easy if done properly - a giraffe sees you coming from a long ways away. You should try to get close - like you would for a dangerous animal - certainly within a hundred yards, and ideally within 50 yards. It will take some time, and you may not get it on the first try, but when you do, you will have a trophy worth of those dreams.