In this paper, the following initial boundary value problem for the one-dimensional stochastic time fractional diffusion equation is considered: \begin{align*} & \partial_t^\alpha u(x,t)- \partial_{xx} u(x,t) = F(t)\dot{W}_x, \quad (x,t) \in (0,1)\times \mathbb{R}_+, \\ & u(x,0) = 0, \quad x \in [0,1], \\ & \partial_x u(0,t) = 0, \quad u(1,t) = 0, \quad t \in \mathbb{R}_+, \end{align*} where \( \partial_t^\alpha \) denotes the Caputo fractional derivative of order \( 0 < \alpha < 1 \) with respect to the variable \( t \), and \( F \) is a deterministic function satisfying \( F(0) = 0 \). In addition, \( W_x \) is the spatial Brownian motion satisfying \( \mathbb{E}[W_xW_y] = \min\{x, y\} \) for \( x,y \in (0,1) \), and \( \dot{W}_x \) denotes the formal derivative of \( W_x \) known as the white noise. The authors deduce the Green’s function for the following equivalent problem in frequency domain: \begin{align*} & \partial_{xx} U(x,\omega) -(\text{i}\omega)^\alpha U(x,\omega) = -\hat{F}(\omega)\dot{W}_x, \quad x \in (0,1), \ \omega \in \mathbb{R}, \\ & \partial_x U(0,\omega) = 0, \quad U(1,\omega) = 0, \quad \omega \in \mathbb{R}, \end{align*} where \( \hat{F} \) denotes the Fourier transform of the zero extention of \( F \) in \( (-\infty, 0) \). This provides the necessary tools for showing well-posedness of the direct problem. Subsequently, the inverse problem is considered which is to reconstruct the diffusion coefficient \( F \) of the random source from the measured data \( u(0,t) \) for \( t > 0 \). It is shown that the modulus \( \vert \hat{F}(\omega) \vert \) is uniquely and unstable determined by the data \( \mathbb{V}[U(0,\omega)] \). The phase retrieval for the inverse problem is also discussed. The paper concludes with some numerical illustrations that make use of a finite difference method to discretize the problem, and in addition a regularized convex optimization scheme is used as a stabilizer.