Let $w(x) > 0$ be a weight function on the interval $[0, 1]$ and let $P_n(x), n = 0,1,2\dots$ be a sequence of orthonormal polynomials of exact degree $n$ which satisfy $$\int_0^1P_n(x)P_m(x)w(x)dx = \delta_{nm}$$
Where $\delta_{nm}$ is the Kronecker Delta.
Let $w(x) = 1$ for $0<x\leq 1$, then first I need to calculate $P_0(x)$ and $P_1(x)$.
I got $P_0(x) = 1$ and $P_1(x) = x - \frac{1}{2}$.
Then I need to find the weights, $$w_j = \int_0^1\prod_{i\neq j}^n \frac{x - x_j}{x_j - x_i} w(x) dx$$
$n = 1$
My issue here is, I am trying to work out $w_1$, but we only have $x_1$ so what does the product equate to?
By convention, the product of an empty set of numbers is $1$. (This is similar to how the sum of an empty set of numbers is $0$). So, $$ w_1 = \int_0^1 1\cdot w(x)\,dx = 1$$ And so, the Gaussian quadrature with one node turns out to be just the Midpoint rule.
By the way, for every degree, the sum of all weights $w_i$ must be $\int_a^b w$, which in your case is $1$. This is another way to see that $w_1=1$. In general, this is a way to check the correctness of your computation of weights.