Quite normal feelings in the first few weeks. Particularly if you have never spent much time away from home\never been rifted morning noon and night\never been so constantly knackered that only your chinstrap is holding you up etc.
Tell him to stick it out - as he gets further into basic he should be fine, not least as self respect and a detrmination not to give up usually kick in by about week 9 or 10. In addition he will likely soon get a weekend leave.
THINK CAREFULLY before jacking in - why did he join? - focus on that.
Agree with Ranger, this day and age we have to respect the potential soldiers feelings. Yes, I must admit I felt homesick after the first few weeks of basic training, and yes i felt like jacking it, but 21 years later I still love it and would not turn back time (feel a song coming on). Anyway most recruits I feel would go through this. Just give him loads of encouragement, if he knows his family is behind him, giving him the support, he most likely will carry on to phase 2.
Ask him why he wants out. If it is something like homesickness or early mornings he will soon get over it. It may be he doesn't like the lads he is with. Again this will pass.
Remember you may be the only link to sanity he thinks he has!
Many young trainees don't like to "go public" on things like home sickness in case they look like wusses to the other lads. Encourage him to seek help from military staff, padre or the WRVS ladies, or what ever they do have.
But try not to give in to what he wants.
If you agree with him that the army is bonk, (even if you have no knowledge of it) then it will be reinforced in his mind. Try to get him to see the fun of it. Like lovely fresh mornings on the drill square, seeing the sun rise at dawn. I used to love that, no really, I did.
Remember he has not seen the army yet. It will get better. He will see what the adverts show. The jollies to hot countries, skiing, sailing, comradeship (this will come).
Agree it is important to find out why he wants to leave first. However his DAOR (discharge as of right) window should open after 28 days (unless things have changed). He can then leave at his own request. This should have been explained to him during the initial admin week.
All but one person in my troop during training wrote an official letter requesting to leave, the one who didn't was a total waster by the way.
I will never forget the moment of relief handing in my letter and feeling I will never have to get up at daft o'clock for PT and inspection. Later that day the troop sgt saw me boot polishing the black bits on the stairs and asked why I asked to jack, he then added that he thought I made a great soldier and had a very promising career ahead of me and we will hear nothing more about this leaving lark.
And that was it, as corny as it sounds that conversation gave me enough confidence to carry on.
Moral is EVERYONE wants to jack it at sometime, I mean one of the guys in my troop was best student, even he had enough