sidedraft webers (which I will assume you mean) are designated DCOE, then the size of the barrels (that's probably not the correct term for webers). Hence they can go 40DCOE, 42mm, 45, 48, and 50mm.
42,48, and 50 arguably aren't as common as 40 and 45 in the mini area. Any of those webers will fit a mini, but it is a question of which is more suited. The bigger the carb on the smaller the engine, the more the carbie has to be "Choked" down in the chokes, and jets.
Typically, I would say 40DCOE are better suited to small bore, and 45mm to 1275cc and above, but a 45mm can be made to work on both. It really depends on what you use the car for and why you want the weber (ie daily driver versus competition car etc).
They are much noisier due to the induction "roar". It can be both quite intoxicating and annoying depending on how long your journey is and how enjoyable the drive is being. Whilst I have had webers on almost all of my cars over the years, I am questioning going to a HIF44 or similar myself which are quieter and can be made to produce similar power. It must be a sign of of old age.
Edit - I would also recommend you talk to a mini expert like Greig Malure at Mini classic, mini King or any of the others as if you stick a weber on a bog std 998 or 1100 you'll most likely just have more noise, no extra power and have a pig of a car to drive in traffic. The specs of the motor (head, cam, exhaust etc) will play a big part. I assume you know you can't easily use the existing exhaust manifold with a weber as it is integral with the inlet manifold - they can be cut, but more suited with extractors.
KB