It's a (primarily British) emphatic way of conveying the improbability/impossibility of something desirable happening (or being true).
In my experience, the fine thing component of the expression doesn't normally serve to inform the other person that the speaker would like [something] to happen, since that's usually obvious to both parties anyway. I understand it as...
It would be better [than it actually is] if there was even a possibility of [something], notwithstanding the improbability of that possibility actually coming to pass.
Usually the implication is that [something] won't happen because of the intransigent of some third party (person, organisation, prevalent attitude, set of rules) that the first two know of, and are aware is either actively hostile or indifferent to the speaker's desires.