I've been working on a discrete version of the "unreliable friend" distribution. It would seem that what I've come up with is equivalent to the following random walk:

Choose $p$ from $U(0,1)$

Start at location $x_0 = a, a \in \mathbb{N} $

Move to $x_{i+1} = x_i - 1$ with probability $p$ and $x_{i+1} = x_i + 1$ with probability $1-p$

What is the distribution on your first arrival at $0$?

Is this a known/explored distribution? Information on r.w. distributions with known $p$ is abundant, but so far I haven't found anything for $p \sim U(0,1)$.

UPDATE: After more research it's clear the distribution is related to the Hitting Time Theorem (alluded to by Jon Peterson below). I've yet to find the discussion of the case where $p$ itself is set by sampling from the uniform. Still looking...