When calling randomPoints, set the strategy to one of these. BruteForce simply tries random points and sees if they are on the variety.
LinearIntersection intersects with an random linear space. Setting the DecompositionStrategy => MultiplicationTable or DecompositionStrategy=>Decompose will change how ideals corresponding to points are broken up into minimal primes which can have a substantial impact on speed. Otherwise, the function chooses which strategy it thinks will be better. See DecompositionStrategy.
Default performs a sequence of different strategies, with successively increasing complexity of the linear subspaces that are intersected.