BMW and Mercedes were recently (last five years) playing with pneumatic and hydraulic valve actuation. The BMW's pneumatic system had a flaw in it, in that, pneumatic systems are on or off - just like a solenoid. The problem they were having is that this placed large amounts of stress on the valves as they slammed home - they were breaking the heads off the valves, etc.
Whereas a cam system, or even the Mercedes Hydraulic system, is gentler. The hydraulic system has a small amount of inbuilt "cushioning", through the liquid. It worked reliably enough, but was held up by costs. With a cam, the valve moves quickly at mid stroke, but slowly at the start and finish.
Another problem was the extra room required above the head for all the hardware - whereas the old OHV setup on a Mini is much more compact. Even a K1100 head would most likely be more compact than pneumatic. I'm not sure how big the solenoids would need to be to move the valve.
You'd also need top notch solenoids - how many cycles would they go through before you had to replace them? Think it through - a valve opens and closes every 2nd revolution. At a lowish 3000rpm, that's 1500 cycles per minute. That's 3000 movements (1 open, 1 close) per minute. That's 50 movements per second, per valve. Now think F1 - 20,000rpm, that's 20,000 movements per minute, that's 333 movements per second!
Here's why I've mentioned F1. I'd imagine your idea is based on the thought that this would be better than a pushrod valvetrain, and a pushrod valvetrain's biggest problem is at high revs. Minis have been raced at over 10,000rpm, they were doing it in the 60's (short stroke 970 'S' motors, somehow driving the dizzy at half the normal speed so the points could keep up! See MiniWorld...). That's still around 166 movements per second. And by that point, you're creating problems elsewhere in the engine, as a long stroke engine like the A Series will have massive linear piston speeds. F1 engines that run at 20,000 rpm have a bore of 98mm. Working the calculations, that means they have a stroke of around 35mm - you can imagine their piston speeds are relatively low compared to their centrifugal speed. Compare that to an A Series, with a stroke between 68 and 84mm. To get an A Series to rev past 10,000, you'll have piston speeds higher than F1. Then think about changing gear at those engine speeds! I think it was Eddie Irvine (driving an old C Type Jaguar at the time) who said that he'd love to have a real gearchange in F1 again, but changing gears at over 12,000 would shatter the drivers' wrists.
F1 is so intriguing - think about how many spark cycles they do per minute. V8 @ 20,000 = 80,000 little fires per minute, 1,333 per second. Wow...
So, yes, your idea is achievable, but it is a huge amount of work (as others have pointed out), and will cause extra stress on the head and valves. You'd need powerful, small, long-lasting solenoids with damping at the point where they "close". And it won't provide a return anywhere near as big as your outlay - the only real advantages you'd have would be in controlling the valve lift... But that's an advantage only maximized when you take out the carbie and throttle bodies and put in fuel injection (preferably directly into the combustion chambers, next to the spark plug, similar to the Alfa Romeo JTS system), then use the valve lift as your throttle (that was the point of BMW and Mercedes' research - it was not because their current valve operation systems (OHC/DOHC) were inefficient). If you can solve these problems...
I'm not meaning to flame you, just to point out the problems. But don't stop thinking - without lateral thinkers, we wouldn't have the Mini.
