The Lithium problem in cosmology was first reported some 30 years ago by Monique and Francois Sprite. The problem that we're having with Lithium is that, in theory, there should be a lot more than we're actually detecting. To put it simply, when we look at the oldest stars in the galaxy, we don't see enough lithium.
But first, an overview of what stars scientists examine and why: The stars that we're observing when we examine this issue are some of the most metal poor stars in the Milky Way, the is true when we look at the Small Magellanic Cloud as well. We look at the most metal poor stars because this is a good way of identifying the older stars in the universe.
As there was no iron created in the big bang, all of the Iron in the universe has been created through the deaths of stars over billions as years. As the universe goes on, more stars die...thus producing more Iron. Population I stars are the newest stars in the universe and have the highest iron content. Population II stars have the lowest Iron content (since little iron existed when they were created) and are the oldest generation of stars in the universe.
When looking at some of the oldest and most metal poor stars, there is a lower content of Lithium-7 than has been predicted by the current big bang models. Various other models and situations have been used to try to explain the Lithium problem, but none of them seem to be able to answer all of the questions.
Studying one bright star in the Small Magellanic Cloud, and the gas/dust that the light has to travel through to get to us, what we have discovered is-- there is as much Lithium-7 detected in the star now as what we believe there should have been shortly after the big bang.
The ultimate problem is that there should be a larger proportion of Lithium-7 at our present time in the universes' history, so the question now isn't just where the Lithium went in the past...but also why there isn't more now.
One of the major problems in trying to resolve this issue is that most models that are able to fix the problem with the lack of Lithium-7 also mess up the abundance levels of several other elements. So trying to fix one problem causes several others. And with many of the changes that are being suggested in the big bang nucleosynthesis to try to account for the lack of Lithium, the two elements that must remain the way that they are after any alterations are Deuterium and Helium.
One study has shown that Lithium-7 can be created within the accretion disks of stellar mass black holes as they take material from their companion star. This model can be used to explain how levels have increased over the billions of years since the big bang, but it doesn't explain the lack in the first place.
We still haven't got any real way of working out where all the Lithium has gone from the early universe, and all of the theories thus far cause other problems. Maybe the James Webb Telescope will be able to shed some more light on the matter in the years ahead of us.