These names are from a list of 6000 names given to Finnish children (aged 1 to 10 years old). Some of the names may seem ordinary to Anglophones, but in Finland they're still considered rare and/or unique.
Habeba
Habso
Haije
Hailey /
HayleyHalla - means "frost" in Finnish
Halla-Lilja - means "frost lily" in Finnish
HanaHanh
Hanna-Elina
HannahHannakaisa
Hanna-Kaisa
Hanna-Liisa
Hanna-Lotta
Hanna-Maaria
Hanna-Maija
Hannamari
Hanna-Mari
Hannamaria
Hanna-Maria
Hanna-Riikka
Hanna-Riina
HanneHanni
Hara
Harini
HarrietHavaHawra
HeddaHedvigHeili
Heini-Emilia
Helea
HeleenHelenHelerin-Melissa
HelgaHeliis
Helinä - means "jingle" in Finnish; also Tinkerbell's name in Finnish
Helja
Heljä
HelkaHelkky - derived from the verb "helkkyä" meaning "to jingle" in Finnish
HellaHelleHelli
Helli-Maria
HelläHelmianna - helmi means "pearl" in Finnish
Helmi-Daniela
Helmi-Elina
Helmi-Elise
Helmi-Elliida
Helmiemilia
Helmi-Helena
Helmi-Ilona
Helmiina / Helmina
Helmi-Kristiina
Helmileena
Helmi-Lilja
Helmi-Lotta
Helmi-Lumia
Helmimaria
Helmi-Maria
Helmi-Matilda
Helva
Helvi
HenaHenna-Kaisa
Henna-Liisa
Henna-Maria
Henna-Riikka
Henni /
HennyHenrica
HenriettaHenriikka /
HenrikaHertta
Hertta-Elina
Heta
Hetaliina
Heta-Maria
Heta-Sofia
Hetastiina
HildaHildaliina
Hilda-Maja
HildeHilde-Maria
HiljaHilja-Amalia
Hiljamari
Hilja-Tuulia
Hilka
Hilkka - means "bonnet" or "hood" in Finnish
Hilla - means "cloudberry" in Finnish
Hilla-Helinä
Hilla-Inkeri
Hilla-Liina
Hilla-Maarit
Hilla-Maija
Hilla-Maria
HilleviHilma
Hilma-Helinä
Hilma-Henriikka
Hilma-Liisa
Hilma-Maria
Hilma-Sofia
Hilpi
Hilpi-Lisbeth
Hilppa
Hilta
Hilva
Himani
Himmu
Hinni
Hippu - means "pinch" or "smidgen" in Finnish
Hkar
Hodan
HollyHopeHoudaHuldaHurija