### Part (i)
1. Definition and Setup:
- Let $A$ be an involution on $V$, meaning $A^2 = I$, where $I$ is the identity operator.
- We need to show that there exists a basis of $V$ consisting of eigenvectors of $A$.

2. **Eigenvalues of $A$**:
- Since $A^2 = I$, the eigenvalues $\lambda$ of $A$ must satisfy $\lambda^2 = 1$.
- Therefore, the possible eigenvalues of $A$ are $\lambda = 1$ and $\lambda = -1$.

3. **Decomposition of $V$**:
- Let $V_1$ be the eigenspace corresponding to the eigenvalue $1$, i.e., $V_1 = \{ v \in V \mid Av = v \}$.
- Let $V_{-1}$ be the eigenspace corresponding to the eigenvalue $-1$, i.e., $V_{-1} = \{ w \in V \mid Aw = -w \}$.

4. Direct Sum of Eigenspaces:
- Any vector $x \in V$ can be written as $x = \frac{x + Ax}{2} + \frac{x - Ax}{2}$.
- Define $v = \frac{x + Ax}{2}$ and $w = \frac{x - Ax}{2}$.
- Notice that $Av = A\left(\frac{x + Ax}{2}\right) = \frac{Ax + A^2x}{2} = \frac{Ax + x}{2} = v$, so $v \in V_1$.
- Similarly, $Aw = A\left(\frac{x - Ax}{2}\right) = \frac{Ax - A^2x}{2} = \frac{Ax - x}{2} = -w$, so $w \in V_{-1}$.
- Therefore, $x = v + w$ with $v \in V_1$ and $w \in V_{-1}$.

5. Basis of Eigenvectors:
- Since $V_1$ and $V_{-1}$ are eigenspaces corresponding to distinct eigenvalues, they are linearly independent.
- The direct sum $V = V_1 \oplus V_{-1}$ spans $V$.
- Hence, there exists a basis of $V$ consisting of eigenvectors of $A$.

### Part (ii)
1. Commuting Involutions:
- We need to find the maximal number of distinct pairwise commuting involutions on $V$.

2. Simultaneous Diagonalization:
- Recall that any set of pairwise commuting, diagonalizable matrices is simultaneously diagonalizable.
- Since involutions are diagonalizable with eigenvalues $\pm 1$, we can assume that all the matrices are diagonal in some common basis.

3. Diagonal Entries:
- In the common basis, each involution $A_i$ can be represented as a diagonal matrix with entries $\pm 1$.
- Each diagonal entry can independently be $1$ or $-1$.

4. Number of Distinct Involutions:
- For an $n$-dimensional vector space $V$, there are $n$ diagonal entries.
- Each entry can be independently $1$ or $-1$, leading to $2^n$ possible distinct diagonal matrices.
- Therefore, the maximal number of distinct pairwise commuting involutions on $V$ is $2^n$.

The final answer is $\boxed{ 2^n }$.