The resilience of a Boolean query is the minimum number of tuples that need to be deleted from the input tables in order to make the query false. A solution to this problem immediately translates into a solution for the more widely known problem of deletion propagation with source-side effects. In this paper, we give several novel results on the hardness of the resilience problem for $\textit{binary conjunctive queries with self-joins}$ (i.e. conjunctive queries with relations of maximal arity 2) with one repeated relation. Unlike in the self-join free case, the concept of triad is not enough to fully characterize the complexity of resilience. We identify new structural properties, namely chains, confluences and permutations, which lead to various $NP$-hardness results. We also give novel involved reductions to network flow to show certain cases are in $P$. Overall, we give a dichotomy result for the restricted setting when one relation is repeated at most 2 times, and we cover many of the cases for 3. Although restricted, our results provide important insights into the problem of self-joins that we hope can help solve the general case of all conjunctive queries with self-joins in the future.