While both of the existing answers touch on relevant matters I believe they both miss the mark in terms of the intended meaning.
@frathoss cites a range of definitions including one which supports the OED definition of 'rubber' as
North American colloquial. To listen (in) on a party telephone line, or on any telephone conversation.
but declares the relevant meaning not to be that one.
@DJClayworth reminds us of the biblical phrase 'the blind see, the lame walk, those with skin diseases are healed, the deaf hear, the dead are raised, and the poor are told the good news' to which the phrases in the short story allude. However, I think reference to 'lame talk' and 'blind rubbers' as noun-phrases is a red herring. Search results for 'blind rubber' mostly lead to 'blind rubber grommets' where rubber is an adjective rather than a noun. Google Books hits for 'lame talk' between 1914 and 1973 primarily brings up either false positives where 'lame' and 'talk' are separated by punctuation, or the O Henry story. Other uses do occur, but they are in the minority and generally post date the 1908 publication of Jeff Peters as a Personal Magnet.
In the story, Jeff Peters is engaging in bombastic tall-talk as part of his spiel to con the Judge, he is aware as he does this that the Judge believes himself to be engaged in a 'con' to bring Jeff Peters to justice. So we can understand the whole bedside performance to be entirely theatrical. Jeff Peters isn't truly trying to sell anything because he knows that the Judge isn't really ill and that the Judge believes himself to be setting Peters up for arrest, and already knows full well that the 'healing through personal magnetism' is fake, just as Peters' bottled medicine is a mere nostrum.
So the spiel is pure performance, only intended to convince the judge that Peters is unsuspecting, and to leave him something to chew on down the line when the con has become apparent:
"'Mr. Mayor,' says I, 'the time will come soon when you'll believe that personal magnetism is a success. And you'll be sure that it succeeded in this case, too.'
So Peters is having fun with the words, using phrases that echo the New Testament but don't convey its meaning, the lame can already talk, the blind can already rubber on their neighbour's telephone conversations. As part of a sales pitch these might be claimed to create plausible deniability that the vendor can't be held responsible for the misunderstandings of his customers. But in this instance he is running a different con and what he has to sell is the idea of himself as an over-confident conman who doesn't realise he is in a trap.
So literally 'rubber' means 'listen in on other's conversations on a party line', but its function is to contribute to bolstering the Judge's belief that Peters is unaware he is being set up.