Heat Capacity at absolute zero As Heat Capacity of solids has two contribution: One arising from lattice, called lattice heat capacity and another from electrons, called electronic heat capacity. At absolute zero, we say that heat capacity of solids is zero, which according to me implies that even a small amount of heat would increase the temperature by finite amount. As at absolute zero, the electrons are in ground state, having no energy levels in the vicinity that they can be excited to, hence adding even small amounts of heat would result in increasing of temperature, hence making the electronic heat capacity zero at absolute zero. What I don't understand is: How the lattice heat capacity is zero at absolute zero?
 A: 
At absolute zero, we say that heat capacity of solids is zero, which according to me implies that even a small amount of heat would increase the temperature by finite amount.

This is not correct. Take expression $CT=H$ where $C$ is specific heat, $T$ is temperature and $H$ is heat energy.
Now at absolute zero, $T=0$ and as you said $C=0$, making the RHS $H$ zero as well. Now if you give a small amount of heat, how can you conclude that $T\ne0$, when we have no idea how $C$ behaves? You can only interpret how $T$ would change depending on how $C(T)$ behaves.
What if you give the system $x$ amount of heat and $C$ is still zero? Since $C=0$, $T$ can take any value including zero and still the equation will hold. Notice a very fascinating phenomena here, when $C=0$ even after you provide energy, this makes LHS zero, which by equality implies the RHS is zero. The Heat taken up by the system is zero even though you just provided some heat!! Now if the system didn't take up heat, definitely $T$ won't change. This strange phenomena does exist and it has to do with Quantum "Quantization" of energy.
To summarize, just because we have a relation $CT=H$ and $C=T=0$, we cant conclude that providing heat to the system will change $T$.

How the lattice heat capacity is zero at absolute zero?

Now, clearly classical theory (Dulong–Petit law) doesn't agree with this. So to understand this, you have to go into various quantum theories like @YoungKindaichi has stated. The intuitive reason as to how this happens is exactly the same reason why electron specific heat is zero (although the reason is inaccurate). Just like you have stated, similar to electrons being in its ground state, lattice vibration/phonon vibration is in ground state too.
Now, to understand why $C=0$, it would be worth to understand what $C$ actually stands for.
$C$ is just a conversion factor, that tells you how much heat energy(which is a form of free energy that can be tapped into) a system has, in terms of the temperature that you can measure. So $C=0$ means that this conversion factor is $0$, that is  the system simply doesn't have any free energy in the form of heat. As a matter of fact the absolute ground state (zero fluctuation) of a system can never have any free energy(in simple terms, energy that can be tapped into). So that is a intuitive way as to why a system should necessarily have $C=0$ in its ground state.

Just put this on firm grounds, what I call free energy here just implies the energy that can be pulled out of the system to do some useful work. So, in terms of standard thermodynamic potential, this would be
$$E:=F-U_0$$ where F is the Helmholtz free energy and $U_0$ is the ground state internal energy(which represents binding energies of various components - which is pretty useless according to me ;p)
A: Near absolute zero, the quantum treatment is necessary. Consider the Einstein solid, Heat capacity plot look like:

The simple explanation for  $C\rightarrow 0$ as $T\rightarrow 0$, because changes in the temperature has no effect on the internal energy when the temperature is so low that only the lower level is occupied and even a small
change in temperature won’t alter that.
