Intriguingly, I have never used references prior to the eighties. The oldest I used were:
R.G. Newton Inverse scattering by a local impurity in a periodic potential in one dimension, JMP 24 (1983)
P. Seba, The generalized point interaction in one dimension, Cz J Phys B 36:667 (1986)
M. Carreau; Four-parameter point-interaction in 1D-quantum systems, J Phys A: Math Gen, v 26 (1993)
There was a book on the topic in 1988, by Albeverio et al, that instead closing the topic, caused some subsequent work due to issues on the denomination of the different solutions. While the plain "delta barrier" was clear to everyone, some different barriers were expected to appear when using the $\delta '$ distribution, and it was controversial if the book had choosen the right solution.
Mathematically the question is about self-adjoint extensions of hermitian operators, and I'd expect it to appear in textbooks addressing this issue.