1. Select a constructor.
2. Click upgrade.
3. Select Avatar as the ship you wish to upgrade to, click ok.
4. From the main screen, select the same constructor again.
5. Click construct.
6. Choose economy starbase.
(The starbase is now built, but the words: "Upgrading: Next Week" will still be there.
7. End your turn.

Next turn, the new starbase you just build will instead upgrade into an Economy Avatar. I didn't notice any differences stat wise and it's not like the ship has the circle of influence that a starbase has, just a different name. I also tried this with a battleship. I also tried it using an influence starbase instead of economy and it put the word Influence in fron of the class name.

Minor bug, but thought I'd pass it along. I did it the first time by accident.