Is there a website or organization cataloging lists of biological species to have been recently found of great economic value? For example, a plant that wasn't of economic relevance a while ago but has been found recently to be important, due to advances in breeding varieties and more crops being planted. The same for species that were already important but have turned to be even more important in the last few years.
I can only answer for plants. The short answer is no, there is no central source of information of that kind.
You are basically asking about the sub-classifications economic botany and ethnobotany, which cover the economy and human uses of plants respectively. Neither field is particularly fast-paced, so you could simply keep up to date with the latest issues of journals like Economic Botany or Ethnobotany.
One key area of development for new valuable crops is in crop wild relatives, but there is no particular publication you should watch for that. Keep an eye on the Crop Trust as they will shortly be releasing, in collaboration with us (the Millennium Seed Bank) some reports and data about this.
If I wanted to keep up to date with all those things over time, I would set up some saved search alerts on Google Scholar for key search terms like 'emerging crop', 'new crop', 'crop wild relative', etc. Just perform a search on Scholar and then click the envelope button to the right of the search field to create a recurring search alert.
One other potential source of information would be to check FAOSTAT each year. You can examine global and local data about area under cultivation, yield, investment, commodity value, etc. of most crops. If something new were to emerge as a big player, it would appear on there.