There is something to know about thumb-shifters: ergonomics.
The better thumbshifters placed the pivot forward of the handlebar, the more affordable ones did not have this.
Here is a pair of Suntour classics from back in the day:
See how the forward placing of the pivot works to place the thumb-shift lever where you need it for the thumb. The other variety lack this subtle detail.
Here are some retro Shimano classics:
These beauties have the option to switch between index and non-index. They are made of top-quality forged alloy parts with no stamped-out tin in sight. The adjusters are particularly satisfactory.
A set of these picked up from e-Bay will grace your BSO and make it have a touch of class as well as be more ergonomic. Both the Shimano and Suntour classics were exceptionally well engineered, for my money I would prefer the lever shape of the Suntour models.
P.S. I introduced 'BSO' to this Q+A site, it is a good-humoured thing. I don't mind BSO's that get actually ridden, however, most of them get land-filled to the detriment of the LBS (Local Bike Shop).
At the moment I am considering getting a Kona Africabike3 - a deluxe BSO. I am also considering getting an actual BSO, however I would have to upgrade the tyres before I took it anywhere.