No, Sol is the name of the sun.
Actually, the Greeks called it Helios back in the day.
The solar system doesn't have a name, it's just called the Solar System.
"Stephen Wolfram is the creator of Mathematica and is widely regarded as the most important innovator in scientific and technical computing today." - Stephen Wolfram
The suns name is just that, "Sun" or as some writers put "Sol".
And this is a definition of "Sun" and "Sol"
sun Pronunciation Key (sn)
n.
often Sun A star that is the basis of the solar system and that sustains life on Earth, being the source of heat and light. It has a mean distance from Earth of about 150 million kilometers (93 million miles) a diameter of approximately 1,390,000 kilometers (864,000 miles) and a mass about 330,000 times that of Earth.
Sol Pronunciation Key (sl, sl)
n.
The sun.
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[Middle English, from Latin sl. See swel- in Indo-European Roots.]