The rennet tablets I have are for large quantities of milk, 50 litres, much larger batches than I would ever make at home. However, separating out a tenth of a tablet is finicky and inaccurate, so I thought it might be better to dissolve half of or a quarter of a tablet in water, and doing the math to add the correct equivalent quantity of rennet solution each time. Can this reconstituted rennet be stored, if so, how and for how long?
Yes, you can. In fact, high-quality rennet is sold in liquid form (and I recommend this over tablets).
Just put the dissolved rennet in a bottle and store it in the fridge (it will degrade at room temperature). It should be good for a few months.
Given your method, though, you'll want to be very sure that the tablet is completely and evenly dissolved.