Is weighted XOR-SAT NP-hard?
A classical result of Berlekamp, McEliece, and van Tilborg shows that the following problem, maximum likelihood decoding, is NP-complete: given a matrix $A$ and a vector $b$ over $\mathbb{F}_2$, and an integer $w$, determine whether there is a solution to $Ax = b$ with Hamming weight at most $w$.

You can reduce this problem to your problem. The system $Ax = b$ is equivalent to the conjunction of equations of the form $x_{i_1} \oplus \cdots \oplus x_{i_m} = \beta$. If $\beta = 1$, this equation is already of the correct form. If $\beta = 0$ then we XOR an extra variable $y$ to the right-hand side, and then we force this variable to be $1$ by adding an extra equation $y = 1$. We define the weights as follows: $y$ has weight $0$, and the $x_1$ have weight $1$. We have now reached an equivalent formulation of maximum likelihood decoding which is an instance of your problem.