What was Inu Yasha's father's name?

Actually.. nobody knows his name... in the 3rd movie he is called Inu no Taisho which means "Lord of the Dogs" :)

Hope I helped!

(By the way, I love the movies and episodes!:D Huge fan!;P) <3

Inu no Taisho

Inu no Taisho (犬の船長 Inu no Taisho?) is an extremely powerful yokai lord who ruled the Western Lands, and was known throughout Japan. Never formally addressed with a name, he fathered two sons by two different mothers: Sesshomaru from his consort, a dog demoness, and InuYasha from a human woman named Izayoi. Inu no Taisho is first mentioned in chapter 12 of the manga as InuYasha's father, "a demon dog that prowled the lands of the west."

Inu no Taisho's major foe in the manga was the dragon Ryukotsusei, whom he could not kill but sealed onto a cliff face with his claw; Myoga calls this battle the "cause" of his death.

The gateway to his tomb is hidden in a black pearl (a mystical gem that creates a path between the mortal and spirit worlds) inside InuYasha's right eye. Within the pearl, their father's body is shown as an enormous dog skeleton in a suit of armor: his true form, according to Myoga. The sword Tetsusaiga, forged from Inu no Taisho's own fang, rests inside this cavernous skeleton, protected by a barrier that repels any non-human with malice toward humans. Because of this, InuYasha inherits the sword rather than Sesshomaru. In the anime, he is voiced by Akio Otsuka. In the English dub, he is voiced by Don Brown.

this is not his real name no one knows his real name(Never formally addressed with a name)

Inutaisho or Inu no Taisho.

I don't believe anyone knows quite for sure.

Inu no Taishou