Yes, Palpatine's ability merely requires the ship to be a friendly ship - so you can modify the dice of any friendly ship on the board, including his own. The new rulebook clarifies that the ship that a special ability is originating from is also a "friendly ship" onto which you can apply that special rule.
In cases where you can't apply abilities to your own ship, you will see the wordage "another friendly ship". For example,
Jan Dodonna can't use his special ability to benefit the ship he is on, but
Jan Ors can. Palpatine uses the latter's wording, and so can modify the dice of the ship he is riding with.
It's also specifically worded to only be usable on friendly ships. You do have cases of card abilities that apply to any ship,
friendly or hostile.