"I'd rather have. . ." is an amazing phrase to me. It can show so many different ideas.
"I'd rather have an apple than an orange." This sentence shows preference of positives. Both are good examples, but given the choice, the speaker would choose the apple.
"I'd rather have dysentery than hepatitis." This sentence shows a reluctant preference. In this case, both examples are awful, but the speaker is in a situation where one of the options must be chosen, so he chooses what he considers to be the lesser of two evils.
If you start a sentence with, "I'd rather have. . .", you should eventually use the word "than". But you don't necessarily need to. Consider, "I'd rather have thinning hair." Nobody wants thinning hair, so we know that the "than" part is not, ". . .than have a luscious head of hair."
What I find interesting about that is that it is actually quite indicative of our culture. Hair is a prized thing, and the more lovely, the more it is appreciated.
Now consider the sentence, "I would rather have a thin beard." I have no idea what the "than" part is. Would he rather have a thin beard than no beard at all? Would he rather have a thin beard than a big bushy one? Would he rather have a thin beard than hepatitis? I just don't know, because I don't now his culture (or his personal beliefs).
I think the cultural aspect of this is the most fascinating part here. You can learn a good amount of a person and their latent beliefs by the things they'd rather have. The more you find out what somebody would rather have, the more you find out who they are and the kind of world they want to live in. It's a great way to solve characterization issues.