Here are Jain baby boy names starting with H, with brief meanings:

• Harishankar (हरिशंकर): Lord Vishnu; one who is like Lord Hari (Vishnu)
• Hemant (हेमंत): Early winter; season of cool, golden mornings
• Himmat (हिम्मत): Courage, bravery
• Harendra (हरेंद्र): Lord of the monkeys? (often used as a name referencing Lord Hari)
• Hitesh (हितेश): Lord of well-being; one who brings good
• Hinay (हिनय): Faithful; loyal (variant spellings exist)
• Harsukh (हर्षukh): Joy and happiness
• Harit (हरित): Green; lordly; one who is green/greenish
• Harshad (हर्षद): One who gives happiness; bringer of joy
• Hemendra (हेमेंद्र): Moon of gold; one with golden mind
• Hiteshar (हितेशर): Lord of welfare; benevolent
• Hiran (हिरण): Deer; precious; noble
• Hemasinh (हेमसिंह): Golden lion; strong and radiant
• Hriday (हृदय): Heart; one with a kind heart
• Harkesh (हार्केश): Lord of the fan/embracer; protector

If you’d like, tell me your preferred meanings (e.g., auspicious, strength, compassion) or whether you want traditional Digambar/Shwetambar variants, and I’ll tailor the list.