How can I change the repository name in my breadcrumbs?

If you're seeing something undesirable like "Islandora Repository" in your breadcrumbs, instead of your repository's name, check the various menus you have enabled -- even the ones you aren't using. The breadcrumb titles come from the menu links that are configured.

Go to Structure -> Menus and check on the links within the various menus. "Navigation" is a likely culprit here. Find any link that points at /islandora, and edit the link title.