Make sure correct replacement pump is used. Check all features against the old pump. Note that with Minis both 3- and 4-bolt pumps exist.
Make sure all ancillary parts are clean. If any part of the engine has been re-machined, wash out block thoroughly to remove any particles of swarf, grinding paste etc. Make sure oil galleries are clean. It is a good idea to drill out the brass plugs and thoroughly clean as sediment and foreign material has a tendency to accumulate in the orifices at the end of the galleries. Do not assume the filter will take care of any dirt. It may have to pass through the pump before even reaching the filter. Dirt may even jam open the relief valve, holding it open and (in extreme cases) cause the engine to seize.
Cleaning parts by wiping them with a cloth is best avoided. Fibers from cloth can accumulate in the oil and end up "matting" on the filter thus restricting oil flow. Dirt can often be transferred from the cloth to parts.
Check all clearances within the engine. Replace all bearings, bushes and seals where necessary. Do not assume a new oil pump will correct low oil pressure – it will not. A pump on its own cannot develop any pressure, it only produces a flow. It is the restriction to that flow, by the engine bearings etc. that creates the pressure. Large bearing clearances mean low oil pressure.
Use the correct gasket (right way around!) when mounting the new pump and make sure all traces of the old gasket are removed from the block. If they are not, the pump will not sit squarely and will result in excessive shaft and rotor wear.
Ensure the slotted tongue (pin drive) is properly engaged before tightening up the mounting bolts.
It is vital that the pump is adequately wetted or primed prior to installation. Pour clean engine oil through the pump and at the same time rotate the drive shaft. If it is likely the engine will not be started for some time, it may be useful to use an oil thickener or oil treatment product (e.g. Vaseline) to prevent run-off and this will assist in "building-up" oil pressure prior to starting.
A worthwhile project is to install a magnetic trap on the gearbox oil galley blanking plate on Minis. This can be easily manufactured using a spare magnetic sump plug. Simply drill a hole in the plate (remove it first!), weld on a large nut with a matching thread and install the sump plug using a copper sealing washer. (Mini Spares amongst others sell them). This will attract most ferrous particles in the oil (which may escape the gearbox strainer) before they enter the pump.
Remove the plug periodically to clean off any metal. Alloy oil pumps and steel or iron particles do not seem to be all that compatible!
As they say Cleanliness is next to Godliness (too bad I'm an atheist). The above procedures seem akin to heart surgery, which I suppose fitting oil pumps on engines essentially is. But that's where the similarity seems to end – high blood pressure is bad for humans, low oil pressure is a worry for Cooper-S owners (but even worse for Jaguar owners!).