The phrase “the battery problem” is shorthand jargon for a problem well-understood by those designing renewable energy systems (especially wind or solar). The problem is that people need power when these renewable sources aren't available. You aren't going to get a lot of solar power at night; this is the most obvious case.
The problem is slightly subtler than that, though. If you look at solar power alone for just a moment, it turns out that you have to account for a lot more than just nighttime. There is less solar power available in bad weather (like a snowstorm) than in good weather. There is less solar power available in the morning and the evening than there is at noon (this is true even for systems that orient themselves to point toward the sun).
For all these situations, what's needed is a way to store power when it is plentiful, for use when it is not. The conventional answer for this storage is a battery, specifically a lead-acid battery very similar to what is in your car. This battery was invented over 100 years ago, and nobody has ever come up with anything that beat its cost vs. storage capacity. But lead-acid batteries are far from ideal; in fact, in many ways they are simply awful. If you've ever carried one of these beasts, you know they are full of environmentally-unfriendly lead. And then there's the nasty acid. Lead-acid batteries used in the way that renewable energy systems need to use them (deep-discharge cycling) wear out quickly, typically in three to five years – at which point they must be replaced. That's an expensive proposition!
Hence the shorthand “the battery problem”. Renewable energy is always going to be of limited value until someone solves the battery problem by inventing a way to store electrical energy that is cheaper, more reliable, and more environmentally friendly than lead-acid batteries.
Here's a Kickstarter project that's taking a stab at the battery problem, using another very old technique (mechanical flywheels) with some very modern twists. Their own claims are for relatively modest improvement over the cost of lead-acid batteries, but with much higher reliability and vastly better environmental friendliness. It's not a quantum leap, but it sure looks like an improvement. They claim to be essentially finished with development other than the need for a better magnetic bearing, which is why they went to Kickstarter for some funding (a very modest amount, actually). I don't know enough about the details of what they need to accomplish to be able to even think about their chances of success...but I wish them well. It's a clever idea, and certainly we could use something better than those danged batteries!
More info (and explanatory videos) here, here, and here: