Multiple explanations, not a single one is universally agreed upon.

"Jesus Christ" has been used as an oath for centuries but Jesus H. Christ is supposedly an (originally) American version, which, according to Roger Smith could have started when ordinary people noticed what looked like IHC/JHC used as a decoration on the vestments of Episcopal clergy. IHC/JHC (earlier and later version, respectively) is what the first three letters of the Greek name for Jesus Christ would look like, even though it's actually iota-eta-sigma.

Another version is that it means Jesus Harold Christ. It stems from the possible mishearing of the Lord's Prayer, which goes "Our Father who art in heaven hallowed be thy name" but could be misinterpreted as "Our Father who art in heaven, Harold be thy name", most likely by children.

One more meaning for the H could be "Holy", as in "Jesus Holy Christ".

Smith, Roger (1994) The H of Jesus H. Christ. American Speech 69:331-335.