Smart clothing on cruise ship for boys? ?

Sounds fine.  If the line has a formal night, then I'd rethink.  My sons weren't nuts of getting dressed up, and both were healthy eaters, so sometimes my wife and I would go to formal night, and the boys ate at the buffet.  Other times, it was an opportunity to teach them how to tie a tie, and they'd wear button up shirts, pants, and decent shoes.  They didn't want to miss out on lobster and prime rib. 

I stopped taking a sports coat on cruises years ago, as I run hot so I'd only wear it until I got to the table anyway.  I felt it took too much too up in my bag, so I stopped bothering.  I'd wear a dark color shirt, black pants, and a darkk tie, so I didn't draw much attention.  I never had any problem. 

More often than not, they would dress as you've stated, polo (tucked in), chinos and dockside shoes.  The host or hostess will look the other way at little kids in shorts, but it's a good opportunity to get them dressed up a bit.  

How old is your son?

Also; shirt and chinos is fine.