Where would be the best place to take the friendship boat into Epcot? Hollywood studios or Boardwalk?

It depends where you are at the time. Both embarkation points lead to the Epcot dock.

If you are staying at beach club, yacht club, swan, dolphin, or boardwalk.... there is a yacht/beach club stop; there is a swan/dolphin stop; and a boardwalk stop for the boats to/from Epcot.... You can pick up the boat at any of these.... as well as at hollywood studios.

Just remember if there is lightning/storms within reach, they stop the boats for safety reasons.

Also remember to check with the boat driver as to which direction they are going -- to hollywood or to epcot -- or your journey will be unexpectedly twice as long!

Also note that the friendship boats do not actually enter the world showcase lagoon. The Epcot boat stop is outside of the International Gateway entrance. The boats that you see offering transportation inside World Showcase Lagoon are different boats that transport guests back and forth across the lagoon to minimize their walking. You may already know that, but I've fielded questions from a lot of folks in the past who were under the impression that they could take a boat from Hollywood Studios to inside World Showcase.