Turbo is actually a good way to go on an MGB. It works well (as does any other form of forced induction) with engines like thtis that have, shall we say, compromised breathing ability. Basically, when you can't get them to suck any more without taking heroic and very expensive measures (ever cost out a full deal SCCA head?) you can just opt to grab the mixture with both hands, as it were, and cram it in there.
And the MGB has all the ports on the same side, which works out pretty well - exhaust out to turbo, and compression side of turbo back into intake. Should be room to do it all.
Problems to solve - sizing of turbo, creating exhaust manifold, creating intake, deciding on suck-through or blow-through system (former is easier as you don't have to pressurise the carb), ancillaries like knock sensor and timing control etc.
You could do some of this by studying a similar kit for a like sized engine and borrowing, but Steve is correct, while supercharging has been developed, you'd be the guy doing the development for a turbo.
Once it has been done though, it should have a market out there. The same sort of pressure (c. 7-8 psi) as used in supercharging would be appropriate.
I would suggest that a kit that used a commonly available carb, probably an SU just to keep it British would be a good idea. I'd nominate either an HS-8 off a Rover 2000TC (early models used the fixed float HD-8 which limits the angle you can mount the carb at while the later HS-8 had a flex jet like an MGB HS8 and is easier to adjust angles, but either would probably work).
I say go for it!