First, we need to define what the little ice age was, and when it was. Keep in mind that it is very difficult to say when it started or when it ended, however we do have enough knowledge to approximate this.
Sometime around the middle of the 1600's, temperatures dropped worldwide. There is little doubt that this was a worldwide phenomena considering the core samples and written accounts that have come from all over the globe, including south of the equator, in Australia, as well as in North America, Asia and Europe.
The change didn't occur suddenly, but was spread over time, which is one reason that it is so hard to determine when it actually started. It clearly had nothing to do with the last glacial period of around 11,000 years ago, but the temperatures dropped precipitously in many places so that summers were cooler and winters were much colder than normal, throughout the globe.
At its peak, the temperatures were so extreme that in London England, they held yearly festivals on the thick ice that formed over the Thames River. It would have been possible at that time to walk from what is now New Jersey to what is now Manhattan, over the ice. Around the world, lakes that never froze in the recorded past, did in regularity. It snowed in Australia. Crops in many areas failed. People died from the frigid temperatures of winter, in places that were normally mild, year around.
What caused it is a bigger mystery. Some scientists have proposed that it was caused by volcanic activity that blocked the rays of the sun. There is a distinct problem with that. Major volcanic eruptions have been occurring since then, and they were before then as well. Yet neither before nor after the little ice age, in recorded history, has there been such a drastic cold change in global temperature, though there have been many major volcanic eruptions around the world.
Others have suggested that ocean currents changed greatly, and since the ocean is the temperature thermostat of the planet, that this caused the change. Yet there is absolutely no evidence that supports such a tremendous change in ocean currents. All evidence is that the currents have remained fairly constant.
The best theory is still missing some pieces. During the worst part of the little ice age, it was observed that the Sun had no sunspots. We now know that sunspots are cause by magnetic vortices on the sun's surface. But we have no idea how the lack of sunspots could cause a lower earth wide temperature.
Still, this idea is compelling, because the sun normally goes through an 11-year cycle, from lots of sunspots, to very few. We don't know what could have caused it to suddenly not have any for 70 years. Again, we don't even know if this had anything to do with the little ice age, but the coincidence that this time of little solar activity coincides with the worst of the little ice age is too much to ignore.
The cause of the little ice age is still much in contention. Yet we do know that it happened, and it happened when there was an unusual lack of sunspots. Could there be a connection? In time, we may learn this. For now, let's just say that there is a very good possibility. The future may hold the ultimate answer.