For any speculation you first have to make up your mind if there was a will or not. Now if you go by probability, an 18 year old girl, that ran away from home is not very likely to already have a will in play, however if they were super loaded, it could've been custom for the family to have all children make a will as soon as they hit 18.
Without a will, the order of inheritance is as follows
1. Spouse and Children, and to keep this super simple, for most cases 50% to each party, meaning even 5 Children would together only inherit 50% with the other 50% going to the spouse.
If the deceased has no spouse or children it's
2. Parents and Silblings.
So as you can see, without a will, the MC ( assuming he is the only child of his mother, which seems very likely ) given the fact that Neil and Lynette were not married would have inherited 100%.
As far as inheritance prodedure goes, nobody gets a box with a 5k$ check, some old clothes, a diary and a letter from a lawyer 2 weeks before they turn 19. So whatever the fuck happened there, the MC didn't inherit shit, either someone send him a box with all the stuff and the "who" is the million dollar question or Neil had it all the time.
That box didn't even belong to any lawyer to begin with, you don't inherit when you reach legal age, you inherit when a person dies, a guardian would have been appointed to manage those assets, and in like all but 2 states that would have defaulted to Neil (assuming he is the MC's biologial father and of course still under the assumption that there is no off setting will.)
And even if Neil would'nt have qualified, he would have known right there and then who was appointed as the guardian. And don't even get me started on the requirements the guardian would have to meet upon accepting guardianship to "act in the best interest of the benefactor" because that sure as shit doesn't mean you throw a 5k check into a box and then seal the box, he/she would have to deposit the money and have the benefactor collect the interests.
It would be a real bummer if DPC just willed that stupid box into existence so the MC has the diary to read and some money to splash. I seriously hope the "who" is relevant and by that i mean BOOM-relevant to the story, otherwise it's a fucked up mess and a cheap tug at the heartstrings.