Here let $\Delta u = 0$ in the unit ball and $$u(1 , \varphi,\theta) = \sin^2 \varphi.$$ What is the value of u at the origin?
So I know that this problem uses green's first identity and I suppose that it uses the mean value property, given that it is asking what is the value of the origin and the average value over a unit sphere is equal to the value at the origin. So I am not sure where to go from here.
Would it perhaps be $$\frac{1}{4\pi} \int^{2\pi}_{0} \int^{\pi}_{0} \sin^2 \varphi~ d\theta ~d\varphi \ ?$$
You'll need to use the right area element for those coordinates: $dS=\sin \theta \, d\theta \, d\phi$, not just $d\theta \, d\phi$. (Possibly with $\phi$ and $\theta$ exchanged, depending on which convention you're using.)
You can tell that something is wrong with your integral since it wouldn't give the average value $1$ if you put $u(1,\phi,\theta)=1$ into it.