What is the meaning of the phrase "chance would be a fine thing"?

It means roughly: "it is unlikely that I should be lucky enough for this to happen, but if it did I would be great."

See Wiktionary.

--- UPDATE ---

As a follow up to why it means that, here is my speculation. It is a simple figure of speech that over time has formed into an idiom. In this case "chance" is used by metonymy to mean the results of chance. The literal meaning would be "If the unlikely event happens". Here an aspect of the event -- its low probability -- is used by metonymy for the event as a whole. If you remove the figure of speech the expression would be "[If] the unlikely event happens, [it] would be a fine thing." The "unlikely event happens" is replaced by metonymy with the word "chance", and we get the original.

After this substitution there are various elements and baggage that attach to the phrase over time to give it the current color of meaning, specifically the strong pessimistic tone, and various ellipses to make the saying pithy.

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.

It's most analogous to the phrase, "If I should be so lucky." Or, more succinctly, "I wish."

Think of "chance" as a stand in for probability. And then add "low" if you like to make the meaning abundantly clear. "The low probability of winning the lottery makes it unlikely to happen, but I would be one happy camper if I did win it." Maybe it all started with the following exchange.

"Want to go in on a lottery ticket together? What do you recon our chances are?" "Chance would be a fine thing!" or, as suggested above the even more pithy "I wish!"