I am trying to find the name of the episode where the Enterprise intercepted a dead person in a package that looked like a cocoon, being sent from a planet to its moon. I can not remember how, but it turned into a person.
During the episode, they talk about life and death. At the end of the episode (after 1 week?), the man wanted to go back to the original intended place and is dead again.
This is the Star Trek Voyager episode Emanations. It's not an episode of The Next Generation, but everything else fits.
The crew of the Voyager are investigating an asteroid and accidentally intercept a corpse that has been 'emanated' from its home planet. The lady they intercepted is revived and spends the week moping, before finally being beamed to her death back on the asteroid.
There is, as you say, considerable discussion of the nature of life and death.
KIM: Yes, but, look, for all I know, your thanatologists are right, and all I saw were the corporeal remains of your people, and you do go on to a higher consciousness.
HATIL: But it's also possible that there is no higher existence for us, that when we die we simply cease to exist.
KIM: I, I really can't say. I don't know what happens to your people after they die. I don't even know what happens to my people after they die.
HATIL: Don't you have thanatologists, people who study death?
KIM: Well, sort of. There have certainly been medical experts, philosophers, theologians who have spent a great deal of time debating what happens after death. But no one's come up with an answer yet.
Not an exact match, but I'm reminded of the Star Trek: Voyager episode Death Wish.
In the episode, Voyager encounters a member of the Q Continuum (immortal and omnipotent beings) who has been imprisoned in a comet (not a moon) for wanting to end his existence. (Not dead, but wanting to die.)
The episode debates topics of life, death, suicide, and the agonizing boredom that immortality eventually brings. It centers around a hearing about whether Voyager should return Q to his people - effectively imprisoning him in solitary confinement for all eternity - or be granted asylum aboard Voyager - at which point he would immediately end his own life.
Representing the Q Continuum during the hearing is the "main" Q from TNG played by John de Lancie and features a cameo by Jonathan Frakes as William Riker, also from TNG. (This could make it be misremembered as a TNG episode.)
The episode ends with Q being granted asylum and is made mortal, but Janeway points out that mortal life is something that he hasn't experienced before, and that he may want to give that a try before ending it all. Despite promising to give it a try, it's not long before he decides to terminally poison himself anyway.
Perhaps this is S4E22, Half a Life.
the Enterprise intercepted a dead person sent from a planet to its moon.
The Enterprise escorts a person sent from their planet to a sun. The goal is research, and the escort is intentional, so not quite an intercept, and not quite a moon. But that's okay, because the person, Timicin, is not quite dead either... His is a culture with planned deaths, and although he is physically alive, he is essentially culturally already dead.
Near the end of the episode, there is some friction between the Enterprise and Timicin's government when Timicin requests to stay aboard the Enterprise beyond his appointed death day; this friction might explain your memory of interception.
During the episode, it talks about life and death.
Most of the memorable dialogue in the episode is Lwaxana's and Timicin's debate about whether he should go through with his culture's traditional death, or experience his life to the fullest extent possible. Timicin's daughter also joins this discussion.
At the end of the episode (after 1 week?), the man is dead again.
Timicin decides to go through with dying at the end of the episode. I don't think the timing is perfectly explicit in the episode, but there are several day/night cycles shown, alternating between Timicin's research and travel during the day and dating Lwaxana after hours, so it was probably a few days from start to finish.
Could this be Season 7, Episode 3: Interface? While it doesn't involve "intercepting" a dead person, it does seem to fit the "talks of life and death" topic. It involves LaForge struggling to accept the reported death of his mother and an entity making use of that to convince LaForge to assist them, while still giving LaForge a chance to cope with and accept his mother's actual death.
LaForge is testing out an interface with his visor that would allow him to remotely control a probe. They are using it to access a ship which has become trapped in a gas giant. During the mission, LaForge receives news that another starship commanded by his mother was lost with all hands presumed dead, but he remains hopeful that she will be found until hard evidence is found otherwise.
However, LaForge later then sees his mother on the original trapped ship via the probe, and his mother claims they are trapped on the planet's surface, having passed by this planet approximately 10 days earlier. Later parts in the episode involve him talking with Data, Troi, etc. about how this could be possible, whether he hallucinated it, and whether this is part of him refusing to accept his mother's death.
Eventually, the crew decides to try and rescue the trapped ship with a tractor beam, pulling it directly out of the gas giant, but LaForge wants to instead pilot the ship lower using the probe to try and stage a rescue mission. He begins this rescue mission without authorization. During the attempt, he learns that what he had seen was not his mother but instead a subspace creature which had become trapped itself along with the original vessel. This creature was somehow able to read LaForge's mind and project the image of his mother because of the probe interface.
LaForge's plan therefore does not find his mother or the survivors of their ship, but it is successful in rescuing these creatures instead. However, the interaction with his "mother", even though faked by the creatures, helped LaForge feel "like I had a chance to say goodbye," seemingly coming to terms with and accepting his mother's death.
Could it be ST:TNG S2E2
Nagilum then states that it wants to know everything about death, asserting absent-mindedly that it would take between a third and a half of the Enterprise's crew to complete its experiments. Picard decides to activate the ship's self-destruct sequence rather than to submit to Nagilum's whims. As the crew prepares for their end, Picard is tested again by Nagilum through peculiar behavior displayed by doppelgangers of Counselor Troi and Lt. Commander Data, both of whom question the self-destruct order.
From the Wikipedia article on ST:TNG S2E2 "Where Silence has Lease"
