Here are some Jain-inspired names starting with H:

• Harshil — joyful, pleasant
• Harit — pure, green, fresh
• Hitesh — one who does good, benevolent
• Hiren — diamond; precious
• Hemant — golden, radiant
• Hiral — gem-like, precious
• Heet — well-wisher, beneficial
• Hruday — heart, compassionate nature
• Hrishik — one with calm and controlled senses
• Hansa — swan; symbol of purity and discrimination

If you want, I can also give:

• Jain boy names with H
• Jain girl names with H
• Modern Jain names with H