So these two are examples of cards where you can exhaust them for an effect based on game state.

We were playing with the GI last night, and the question arose: If my ship with him on it has already activated, can I still trigger his effect later on it that turn?

We didn't inherently see anything wrong with this but I realise there are other cards with potentially similar scenarios - like Strategic Adviser.

Is there some clarity re this?

In Grand Inquisitor's case is very clear, it only checks if an enemy ship at distance 1-5, you (the ship The Inquisitor is) may change it's speed. There is no difference wheather that ship has already activated or not.
The difference with Strategic Adviser is the parte of this last one which says "When it is your turn to activate" where it can be argued that it is referring to the turn where the ship Strategic Adviser is. Meaning that you can use that card only if that ship can be activated.

There is no requirement to be able to activate on the grand inquisitor... and indeed on cards in general (re: Interdictor and targeting scrambler)

the wuestion is only on Strategic Advisor due to its wording.

15 hours ago, Jambo75 said:

Is there some clarity re this?

For strategic adviser there is. It is over here . It was even on the same page.

Well, it missnig a clear answer. Because there is non.