Fr. Z has wrote a fictional scenerio on the process of merging two or more parishes into one parish. While the story (which can be accessed here) is hilarious and unorthodox, the very same processes and ideas on how a parish is named or renamed is one of the sad realities of the Post-Conciliar period.
First of all, it is the bishop of the diocese who determines what the names of the parish churches are. For example, if a church building that burned down was St. Mary's, and on the day of consecration, the bishop decides to consecrate it as "Our Lady of Victory", the change of the parish name is effective immediately. If the old St. Joseph's Parish church building had relics of St. Therese in a side altar, and they are transferred in the new high altar, the parish is now known as "St. Therese Parish."
Now, here comes the point that Fr. Z is trying to make. If the bishop decides that St. Peter's and St. Paul's are to be merged into one parish, and that they use St. Peter's Church, then the bishop has the authority to change the name of the actual parish, but not the parish church. In the Pre-Vatican II days, this parish church would not be renamed Ss. Peter & Paul unless (1) the church is reconsecrated, with the relics of BOTH St. Peter and St. Paul placed in the high altar, or (2) the Holy See changes the name of the parish church, which is rarely thought of.
So while the names of the actual church buildings would survive a parish merger, the acutal names of the parishes could be changed by the bishop.
But sadly, many bishops leave the naming rights up to the parish council, which I was a member of one once, sad to say. They could name the parish after a saint who's relics are not in the mensa, or even worse, some profane name, such as "Peace Parish", "Rainbow of God", "First Catholic Church of Anytown", or even "Satan Hall." Better yet, they could "sell" the naming rights as if it were a sports arena, such as "Liberty Church", "Wegman's Worship Center", or "Prudential House of God." (Hey, I still prefer "New Boston Garden" over "TD Garden", since everyone still calls it "The Garden.")
The bottom line is, that bishops have to be bishops. They cannot build new "faith communities" but churches, where the Blessed Sacrament is located and not a peep is heard. A place where people dress with dignity, not like strip club material you see in most Novus Ordo houses of worship today.