Following the computation of the THH (topological Hochschild homology) of $\mathbb{F}_p$ as outlined in Krause-Nikolaus. 
We use the fact that $\mathbb{F}_p$ is initial $E_2$ ring with $0=p$ to compute

$$\mathbb{F}_p \otimes_{\mathbb{S}} \mathbb{F}_p \cong \mathbb{F}_p[{\Omega^2 S^3}]$$

Then, 

$$THC(\mathbb{F}_p) \cong \mathrm{Hom}_{\mathbb{F}_p[{\Omega^2 S^3}]}(\mathbb{F}_p,\mathbb{F}_p)$$

Now using 

$$\mathrm{colim}_{BG} {G} = \{ \star\}$$

We have that 

$$\mathrm{colim}_{\Omega S^3} {\mathbb{F}_p[{\Omega^2 S^3}]} = \mathbb{F}_p$$

and hence,

$$THC(\mathbb{F}_p) \cong \lim_{\Omega S^3}{\mathbb{F}_p} \cong \mathbb{F}_p^{\Omega^2 S^3}$$

where the action is trivial because $\mathbb{F}_p$ is discrete. Assuming the above is correct, I now am not sure how to compute this.