Here are some names for a private boarding school, categorized for different desired images:

Classic & Traditional:

• Ashworth Academy
• Blackwood Manor School
• Charterhouse Preparatory
• Cranbrook Hall
• Elmwood Academy
• Fairfield College
• Greystone School for Boys/Girls
• Kingswood Academy
• Oakhaven School
• St. Augustine's Preparatory School
• The Barrington School
• The Hawthorne Institute
• The Meadows Academy
• Wellington School

Modern & Progressive:

• Apex Academy
• Brightwood School
• Catalyst Academy
• Innovation School
• Meridian Academy
• Pinnacle School
• Summit Academy
• Synergy School
• The Ascent School
• The Horizon School
• The Innovation Institute
• The Vanguard School

Rustic & Nature-Inspired:

• Birchwood Academy
• Cedarwood School
• Glenwood School
• Meadowbrook Academy
• Riverbend School
• Stonebridge Academy
• The Highlands School
• The Redwood School
• Willow Creek Academy

Unique & Evocative:

• Aethelred Academy (evokes a sense of history and mystery)
• Aurelian School (suggests gold and excellence)
• Bellwether Academy (implies leadership and innovation)
• Caerleon School (a Welsh name suggesting strength and heritage)
• Eldoria Academy (sounds fantastical and aspirational)
• Luminary Academy (suggests brilliance and enlightenment)
• Solstice Academy (evokes a sense of change and renewal)
• Veritas Academy (Latin for "truth," suggesting integrity)

Location-Specific (Adapt to your actual location):

• [Town/County] Academy
• [Local Landmark] School
• [Nearby River/Mountain] School

Remember to check for trademark conflicts before settling on a name. Consider also adding a tagline to further define the school's identity.