The prayer Frances provided is called the Jesus Prayer or the Publican’s Prayer or the Prayer of the Publican (which can be found in Luke 18:13). I would understand praying with the holy name of Jesus to refer to meditating upon Jesus’ very name. It is usually considered a more advanced prayer because one must be rightly formed in theology and spirituality for the name of Jesus to evoke the correct prayerful state, carrying behind it the full force of Jesus’ saving life, death, and resurrection.
Here is a website which might interest you about the usage of the Holy Name of Jesus in Roman Catholic prayer history. Here is a Litany to the Most Holy Name of Jesus, as well.
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