BB guns generally tend to be relatively inaccurate.
At 7YO, and with him and the family having an interest in shooting.
With a BB gun, he might lose interest, due to that lack of accuracy.
Air rifle(pellet) - A couple recommendations.
I suggest going on Youtube and watching reviews of each.
Both are multi-pump.
Crosman 1077 / 1077W
Is a low power CO2, and the "W" version has a wood stock.
Daisy Avanti 853
A single pump target rifle. Can be purchased(refurbished) through the
CMP for around $100.
"blowhard who works for an airgun retailer"
A person who blusters and boasts in an unpleasant way.
"Works for an airgun retailer"
I own a small motel in Branson Missouri, and have never worked for
an airgun retailer/manufacturer/etc
I also agree with those suggesting a .22 rimfire, they make a great, first rifle.