You can choose: both versions exist!
a) You can take the product $\check C^n(\mathcal U,\mathcal F)=\!\!\prod_{(i_o,\ldots,i_n)}\!\!\mathcal F(U_{i_0,\ldots,i_n})$ over all $n+1$-tuples, so that indeed there will be much redundancy in your groups i.e. they will be repeated.
b) Or you can put some total order on $I$ and consider the complex $ \check C'^n(\mathcal U,\mathcal F):=\!\!\prod_{i_o\lt\ldots\lt i_n}\!\!\mathcal F(U_{i_0,\ldots,i_n}) $
which is clearly more economical.
b') There is a variant where you use the subcomplex $\check C_{alt}^\bullet(\mathcal U,\mathcal F)\subset \check C^n(\mathcal U,\mathcal F)$ of the complex in a) consisting of alternate families: for example if $n=1$ you require that $s_{ij}=-s_{ji}\in \mathcal F(U_i\cap U_j) \;\text {for } \;i\neq j$ and $s_{ii}=0.$
These complexes give the same cohomology groups: it is pretty clear for b) and b') and it requires a calculation to show that the inclusion of complexes $\check C_{alt}^\bullet(\mathcal U,\mathcal F)\hookrightarrow \check C^\bullet(\mathcal U,\mathcal F)$ yields isomorphisms at the level of cohomology groups $ \check H_{alt }^n(\mathcal U,\mathcal F)\hookrightarrow \check H^n(\mathcal U,\mathcal F) $
In that generality I must admit that all this is a bit boring.
For familiarizing oneself with effective calculations in low degrees, I recommend §12 of Forster's Lectures on Riemann Surfaces in which he very explicitly computes, for example, the first Čech cohomology group $\check H^1$ of some sheaves on Riemann surfaces.
Edit
Let me say, as an answer to Galoisfan's question, that version b) is much more powerful.
Here is an example:
Let $X$ be a Riemann surface and $\mathcal F$ a coherent sheaf.
If $\mathcal U=\lbrace U_0,U_1\rbrace$ is a covering of $X$ consisting of two open sets, then obviously $\check C'^n(\mathcal U,\mathcal F)=0$ for $n\geq 2$ since you cannot extract a sequence ot three strictly increasing numbers from $\lbrace 0,1\rbrace$ !
But if $U_0,U_1\subsetneq X$ are strict open subsets, they are Stein and Leray's theorem says that $ \check H^n(\mathcal U,\mathcal F) = \check H^n(X,\mathcal F) $, the genuine cohomology of $\mathcal F$ (i.e. the inductive limit over the coverings of $X$).
So version b) lets you prove that all genuine cohomology groups $\check H^n(X,\mathcal F) \; (n\geq 2)$ are zero: quite a remarkable theorem!
In the same vein, for every algebraic variety $X$ that can be covered by $n+1$ open affine subsets ($\mathbb P^n$ for example ) and every coherent algebraic sheaf $\mathcal F$ on $X$, the totally ordered version of Čech cohomology shows that $\check H^k(X,\mathcal F) =0$ for $k\gt n$.