If he has a 5 on his card, then rolls a 1 for leadership, can he replace the 1 with the 5, then, since he doesn't have a d6 on his card, put the 1 on his card?

No, for two reasons. You're choosing to replace the "1", so it's no longer there later to place on his card. And the first part won't trigger if there's already a d6 on his card.

Wait, re-read my question. I'm pretty sure the first part works:
Start of turn:
Card: 5

Then he rolls Leadership:
Roll: 1

According to the second part:
"When a character within range rolls an attack roll or rolls for Blades/Claws/Fangs or Leadership, you may remove the d6 from this card and replace a die in the roll with it."

So the roll is replaced and is now a 5 (successful leadership).

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The "real" question would be if the first part of his power can kick in, since he just used Leadership:
"When Mirror Jean-Luc Picard uses it, if there is no d6 on this card, you may place the d6 result on this card."

which was a "1".

The roll has been replaced with a 5. There's no roll of 1 to place on the card anymore. The question is, Can jean-Luc keep that 5 he just used and place it on his card again? Essentially abusing Leadership indefinitely?

So the critical aspect of the question is: What comes first, timing wise?

a) He uses Leadership
or
b) He rolls for Leadership

Are those triggers simultaneous, so the active player can arrange their effects, or does one always come before the other?
I believe a) always happens first, which would mean the time to place a d6 on the card has come and gone by the time you remove the current d6 from the card.

I believe. When you use Leadership, you check to see if there is a D6 on the card or not. If there is, you may NOT put the result of the leadership roll on the card. So either;

a) you have a 5 on the card, and you can replace a roll of one, but not put the result on the card

or b) you have nothing on the card, you roll a whatever, and place the result on your card.

Which would mean he can only place a "rolled" die, not a replaced one.

Correlation does not equal causation. If we’re right, and he can’t keep regaining the same d6, then it’s because of the timing of his two triggers, not because one d6 is rolled and another isn’t. If someone else replaced the d6 in his Leadership roll (another Picard for example), he should have no problem grabbing that one when his own card is empty.

You're right, I had to recheck the wording. Is this the sort of thing that gets errata'd or will we get an official ruling?

As far as I can tell, there’s nothing wrong with the current wording. I wouldn’t mind getting confirmation from one of the more experienced rules forum guys, but otherwise, it should be fine as is.

The triggers are not simultaneous. It doesn't work.

This.

In this scenario:
Declare using leadership: check for d6, there is a d6 on the card, it does not trigger.

You can then replace to make the leadership roll successful, but you will not get to place it on the card, because when leadership was checked on use there was a d6.

Thank you. Nice to get it confirmed (again)