You can respond to another event, like your opponent playing a spell or activating an ability.
So no, there is no opportunity to respond to your spell once it has begun resolving. You will get priority after the spell resolves, and at that time you can activate your Lion's Eye Diamond.
See this question for more details: Can a player respond while a spell is resolving?
To achieve what you're wanting, you'll have to do as follows:
1) Cast Gitaxian Probe
2) Before passing priority to your opponent, activate Lion's Eye Diamond
3) Once you discard your hand and get your mana, Gitaxian Probe will remain on the stack (and can resolve as normal).
The important thing is to remember to immediately activate Lion's Eye as soon as you cast Gitaxian Probe (BEFORE passing priority to your opponent, otherwise it will be too late (unless he responds to gitaxian probe).
Not as you have written, while Gitaxian Probe is resolving no one can do anything since no one has priority.
However you could cast Gitaxian Probe, hold priority, and then activate Lion's Eye Diamond, thus you will get to draw after discarding your hand because Gitaxian Probe has not resolved yet (as you did not pass priority). This won't allow you to look at your opponent's hand before deciding of course.