French Meaning: The name Delaney is a French baby name. In French the meaning of the name Delaney is: From the elder tree grove. Irish Meaning: The name Delaney is an Irish baby name.In Irish the meaning of the name Delaney is: Competitor's child; from the river Slaney.n French the meaning of the name Delaney is: From the elder tree grove. Irish Meaning: The name Delaney is an Irish baby name. In Irish the meaning of the name Delaney is: Competitor's child; from the river Slaney.