Here are Jain girl names starting with H, with simple meanings:

• Harini — deer or doe (a gentle, auspicious animal often symbolizing grace)
• Hemangi — golden-bodied; golden or shining one
• Hemlata — gold (hema) + vine/creeper (lata); conveys golden beauty
• Hema — gold; precious
• Hiranmayi — made of gold or gold-like; precious, shimmering
• Himanshi — born of snow or ice; pure, cool

If you’d like, tell me whether you want names that are more strictly from Jain scriptures or more commonly used in Jain communities, and I can tailor the list.