Longish Post. Beware.
It used to be that the majority of minor league teams were named after the team they were affiliated with. There were, of course, historical teams that eventually affiliated with clubs (The vast majority of the old independent PCL eventually became affiliated with MLB teams but kept their historic names and on the East Coast, a team like the Toledo Mud Hens would fit the same bill).
But, when the Red Sox moved their AA team from Pittsfield, Mass. to Pawtucket, RI in 1970, the six team Eastern League had five affiliated teams, all named after their affiliates (Pirates, Phillies, Senators, Red Sox, and Yankees). The Red Sox also had a team in the Western Carolinas League named the Red Sox (in Greenville, by the way)... where all six teams were affiliated and named after their affiliates. Their Carolina League Team (Winston-Salem) was also named the Red Sox (where four of the seven affiliated teams had the same name). FSL, same thing (9 of 11 affiliates). The only exception to the rules for the Red Sox in 1970 were the NYPL where they went by the Jamestown Falcons (they were one of three exceptions in an 8 team league... the next season they would move to Williamsport and become the Red Sox) and their IL affiliate in Louisville (Colonels... only one of eight teams in the IL were named for the affiliate (Richmond Braves)).
When Louisville move to Pawtucket in 1973 (with the ensuing move of the PawSox to Bristol)... all the Red Sox affiliates were named Red Sox... with the exception of the New York-Penn League, where they moved to Elmira (and had a long association with that club) and were the Pioneers.
[The NYPL affiliates from 73 on were the Elmira Pioneers (73-92), the Utica Blue Sox (93-95 - The Blue Sox name had nothing to do with the Red Sox... they changed their names when they became independent in 1981 after the Blue Jays dropped them) and then the Lowell Spinners (96-current).
Now, again, the use of the Red Sox name in affiliations was not unique to Boston throughout this period. For example, in 1986 (just a random year), the International League... full of historical teams (Tidewater Tides, Columbus Clippers, Toledo, Syracuse Chiefs, Rochester Red Wings) had only two teams named by affiliation (PawSox and Richmond Braves)... but the Eastern League had all eight clubs named as their affiliate. After ending their affiliation in Winston-Salem in 1984 (where for the only season with the Red Sox, the team was not named the Red Sox (Winston-Salem Spirit)), the team move onto Greensboro where they kept the name of the Hornets. However, the Hornets were one of only three teams in the ten team league not named for the affiliate club. And in the Florida State League, nine of the eleven affiliated teams were named after their MLB club. Even the New York-Penn League, the majority of clubs were named for their parent; nine of the twelve.
But, things started to change. When the Red Sox left New Britain and became the Trenton Thunder (1995), it was the first time their AA team was not named the Red Sox since 1958, when they had a Southern Association team (Memphis Chicksaws). 1995 also saw the Red Sox move from the Carolina League to the MidWest League with their move from the Lynchburg Red Sox to the Michigan Battle Cats. Their A-Ball club has not been the Red Sox since (Augusta Greenjackets, Greenville Bombers/Drive). Even their High-A Club had a few seasons not as the Red Sox (Wilmington Blue Rock and Lancaster Jet Hawks).
In 2018, the landscape has changed. Teams are not named after their affiliation, they are their own entity. The only team in the IL named after their parent club... the Pawtucket Red Sox. [Gwinnett changed from the Braves to the Stripers this season, the Indianapolis Indians are not affiliated with Cleveland.]
The same is true when you look at all the non-complex leagues (aka not Arizona, Gulf Coast, Dominican Summer) with the exception of the Appalachian League (where all 10 clubs are named for their parent organization). In the 16-team PCL, only the Oklahoma City Dodgers have the name of their parent club. In the EL... only the Reading Fightin' Phils have a passing resemblance to their parent club with the Binghamton Mets becoming the Rumble Ponies. Southern League (10 team), the Mississippi Braves are the exception. Texas League (8 teams), the exception to the rule is the Springfield Cardinals. California League (8 teams), the exception if the San Jose Giants. In the Carolina League, there are three exceptions (Salem Red Sox, Potomac Nationals, and the Buies Creek Astros)... but that will be reduced by one when Buies Creek moves into their permanent home in Fayetteville. The FSL, which was historically named for parent clubs, have only four of 12 with affiliate names after the Tampa Yankees went to the more historic Tampa Tarpons name [the four are St. Lucie Mets, Palm Beach Cardinals, Dunedin Blue Jays, and Lakeland Flying Tigers]. In the 16-team Midwest League, the South Bend Cubs are the exception. The Rome Braves stand alone in the South Atlantic. The Staten Island Yankees and Connecticut Tigers are the exceptions in the NYPL. There are no exceptions in the Northwest League as the Spokane Indians are not affiliated with Cleveland. The Grand Junction Rockies are the exception in the Pioneer League.
I would never change the name of the Pawtucket Red Sox to something else... they have been the PawSox for nearly 50 years. That is historically significant. However, when they move to Worcester... it's a new club with new history. They don't need to be the Red Sox. And... my guess is that Larry and the Dentist are shrewd enough to know this and the Worcester club will, in fact, not be named the Red Sox.
But, I've been wrong before.