I follow the Tom of Holland blog which is about mending and repair, mostly of knitting. His latest post is called Mending books and it is books about mending, and is well worth reading. There are some lovely examples included. I have not yet seen libraries offering mending programs. It does not mean they are not happening, I have not seen them and would be interested if libraries are offering them - so please add a comment if you know of one.
The mending and repair would fit will with sustainability, and also seems a very neat fit for libraries. Most public libraries will have some collection items relating to mending and repair and these collection items could be combined with library programming. It is a nice way to promote the collection, encourage repair, and help with sustainability. It can also be used as a way to show people how to search on Trove, and do text correction, which is a kind of mending.
There could be partnerships in the programming, like the one on upholstery running as part of the Sydney Design. Partnerships with Mens Sheds could also be valuable, and some libraries are doing work in this area. Or you could establish something like the Brighton Repair Cafe, or work with partners to do this.
It would also be a fit with some of the maker activities running in libraries. It is not making something new, but it is making something older be still usable, rather than becoming discarded, and it can add new character (see this example of mending on the Tom of Holland blog).
Even if you will never mend something, read the Tom of Holland blog for enjoyment.