This Feast is associated with the Feast of the Circumcision, for it is when a child was circumcized that he received a name and was accepted as a son of Abraham and a full member of his family (the Feast of the Holy Family will come soon, after the Feast of the Epiphany on the 6th). So honored is His Holy Name that devout Catholics bow their heads (men removing their hats) at the sound of "Jesus" (or "Iesus" in Latin), both inside and outside of the liturgy. To protect the sacredness and honor due the Holy Name, when hearing the Name of the Lord taken in vain, it is right to pray "Sit nomen Dómini benedíctum!" ( "Blessed be the Name of the Lord"), to which the reply, if overheard, is "Ex hoc nunc, et usque in sæculum!" ("from this time forth for evermore!"). .continue at Fisheaters
While there are no special customs that I am aware of, it is so that if we utter His Name with devotion, we receive a partial indulgence. A plenary indulgence, under the usual conditions, may be received if we visit a church or chapel on this day. A good thing to do to fulfill these conditions is to recite the Litany of the Holy Name, especially in a church or chapel. You can download the Litany, in Microsoft Word .doc format, in English and Latin.
And here are the thoughts of St. Bernard of Clairvaux (who wrote the hymn "Iesu Dulcis Memoria" which is sung today) on the most holy Name of Jesus.