"The mining of rare-earth minerals was at one time dominated by the United States. The People's Republic of China has since come to dominate the market."
If it'd be pure economics, then yes. But Japan and China are not necessary the best friends, so there is a strong geopolitical factor at play. China has the (quasi) monopoly and China with strategic embargo/tariffs/creative strings attached could easily grind the tech manufacturing in Japan to halt, impacting it's economy severely. This is not theoretical, they already impacted Japanese economics this way.
So now Japanese politics has a strong incentive to change the situation, creating a second source. It might be more expensive as long as the supply lines are still open, but it's a strong hedge against politically motivated supply line disruptions.
Simply said, exploiting these deposits even if they are more expensive than the market price can be seen as an insurance. It also strengthens Japan's negotiating position when their industry can't be held hostage so easily.
ps.: Also keep in mind, that it takes considerable time (years) to develop a mining and refining operation, so they can't just wait until a disruption occurs before starting with the mining.
https://en.wikipedia.org/wiki/Rare-earth_mineral
Presumably because China doesn't mind the cost to its miners or the environment (for now).
If that should change then it might start to be viable to mine this deposit.