I don't think it is greatly important to have a school name, although it has helped us in different small ways.
BUT-- having a mission statement has been foundational. We are committed to homeschool all of our children, without government help, to the end. A mission statement reminds us of our goal.
We chose a mission statement AND a bible verse. From there we decided on a name. We also took into consideration our strong beliefs and any family or area characteristics. We live in Oregon, so it is very green, rainy, flowery, mountainous and has a lot of creeks and rivers. We are also "spirit-filled".
Here is what we have posted on our door:
Our mission in homeschooling is...
to raise our children to desire a deep relationship with the Lord, to be spiritually wise, knowledgeable and well skilled in areas that will contribute to their personal life mission, and develop vision, godly character and compassion for the lost- in the comfort and protection of their own home.
"...like a tree that is planted near water.
It sends out it's roots beside a river.
It is not afraid when heat comes. It's leaves are always green
It does not worry when there is no rain.
It always bears fruit."
Our homeschool name is Riverside Bible Academy and we feel that this name reflects the mission and scripture we are using.