The judicial system is, at heart, a method of resolving disputes. Sometimes those are disputes between civilians (civil suits) and sometimes they are criminal cases, disputes between people and the state.
The most obvious and easy place to start is with small claims courts. Commercial arbitration handles many disputes that would otherwise end up in small claims courts, but we don't exploit this anywhere near enough. Most people just rely on their bank to act as a dispute mediator via the credit card chargeback mechanism, but this is a one-size-fits-all solution and banks are often not good at mediating disputes. There's lots of fraud and problematic outcomes.
The place where most of the better-law-through-tech research is happening right now is the Bitcoin community, because of the general focus on decentralisation, global trade and frequent desire to avoid relying on government. So we have for example BitRated which is a platform for doing dispute mediated Bitcoin transactions, where anyone can be the dispute mediator. So you can get a fluid, international market of specialised judges who are experts in very particular types of transactions, like software contracts etc where "I didn't get software of sufficient quality" is not a dispute that makes sense to handle via a chargeback. And it can all happen over the internet.
That's a very simple example. More complex examples involve specifying a contract in the form of a computer program and then effectively having the program be the "judge". I wrote about how to implement this, again with Bitcoin, several years ago. The technology is not that complicated actually. The hard part is figuring out the right user interfaces to make it easy. Presumably only very simple and precise contracts could be managed that way, so there's still open research in how to craft these digital contracts such that you can escape back to human judgement if there's an exceptional case.
When it comes to criminal rather than commercial cases, probably the best way to apply technology to reduce costs is to allow remote lawyering. That is, you should be able to outsource your legal representation to someone who isn't physically present. They may be rather good and experienced, but just lives out in rural areas or in a country where the cost of living is cheaper. The issue here is not really technical but rather just institutional inertia.
The UK is putting its judicial system under tremendous financial pressure at the moment, to the extent that some criminal cases are just being abandoned because there's insufficient money to run them. They're (finally!) starting to experiment with allowing small claims court cases to be resolved over the phone, and also looking at decriminalising TV license violations to reduce pressure on the system. But you get the idea - the judicial system innovates extremely slowly even when being sliced to the bone. So don't hold your breath.