Some good private school names are Parkside Preparatory School, St. Ambrose's Academy, Marie Curie Institute, and the School of Learning and Inquiry. Good name ideas can come from place names, from the names of famous people or those of the school's founders, and from words or ideas that evoke something important about the school.
Place names can come from the town where the school is located, the street it's on or a local landmark. Examples are Andorra Private Academy, Oak Street School and Mountain Lake Prep.
Naming a school after someone who exemplifies the school's philosophy or goals can set the tone for it; The Kipling School, Steve Jobs STEM Prep and Louisa May Alcott School are examples. One family mentioned on A2ZHomeschooling.com named their school the Ben Franklin School of Discovery. The namesake can honor a family member or even immortalize the family dog, such as Grover Day School.
Acronyms may make good school names. MidwestHomeSchoolers.com gives the example of AFEA Academy -- AFEA stands for Awesomest Family Ever Around, but only insiders know what the initials mean.
An important consideration is that the name of a real private school or home school goes on the students' transcripts for other schools and colleges. This may be a reason to rule out offensive or excessively silly names. Huggy Bear Kindercare and Disney Princess School for Girls may be fine choices for preschool and elementary years, but they are perhaps too whimsical for higher grades.