Traditionally almost all amateur contributions to astronomy came from observational hobbyists finding comets and supernovae, which benefit from having more eyes on the sky and don't require the absolute largest telescopes. Most of the sky goes unobserved each night, so just looking for transient phenomena often pays off. People like Joseph Brimacombe are well known to supernova observers, since they often are the first to see new supernovae. Note though that these people often invest significant personal resources into their hobby, and it's not clear what will be left for amateurs once the next generation of survey telescopes is built.
In terms of data and code, I've heard of some hobbyists helping astronomers find exoplanets. This is less solo work than the observations -- these people are often formally or informally members of research teams. There are all sorts of clever data reduction techniques being developed these days, but do note they are not generally pure abstract data analysis -- much astrophysical insight goes into the data pipelines these days.
Zooniverse was already mentioned. This is a collection of "citizen-science" projects, and it has more than its share of astronomy. This is no surprise since astronomy has enormous datasets, and in fact Zooniverse grew out of a single galaxy classification project. These projects are a nice way to get your feet wet with various data reduction projects, but note their focus in on crowd-sourcing results (usually to develop training sets for their own algorithms, which eventually replace the crowd-sourcing). If you want to actually develop algorithms and apply them to large datasets, you can try getting in touch with the people behind a particular crowd-sourced project and ask for more data to work with on your own. While sometimes the latest astronomy data is kept private for a time (no one wants to spend all the time and effort getting data only to have someone else swoop in and publish all the results), much of it is public or sharable.