Lower bound for the number of lattice points on high dimensional spheres

It is easy to see that no integer of the form $8k+7$ can be written as the sum of three squares. So there can be no universal lower bound for $N(r,3)$ that is better than $0$. (Indeed the integers not expressible as the sum of three squares have an exact characterization.)

For $d\ge5$ at least, the circle method as applied to Waring's problem gives essentially an asymptotic formula for $N(r,d)$; more precisely, it gives an asymptotic order of magnitude term times a "singular series" leading constant depending on arithmetic properties of $r^2$, but that leading constant is bounded between two universal positive constants.

My answer to this MO question contains the answer to your question, especially if you take into account that $L\left(1,\left(\frac{D}{\cdot}\right)\right)$ can be estimated unconditionally (i.e. without GRH): $$|D|^{-\varepsilon}\ll_\varepsilon L\left(1,\left(\tfrac{D}{\cdot}\right)\right)\ll \log|D|.$$ The lower bound is ineffective (i.e. we don't know the implied constant), and it is due to Siegel (1934). The upper bound is effective and older. Both bounds are explained in Montgomery-Vaughan: Multiplicative number theory I.