Best place to start is take a look at the Getting Involved page on our development and planning wiki:
http://wiki.jqueryui.com/Getting-InvolvedYou'll find links there to our coding standards, where we track issues, as well as our source code, how to work on issues and submit fixes.
You can also find open items on our Roadmap, or at least see what doesn't have a name next to it:
http://wiki.jqueryui.com/RoadmapFixing bugsSome plugins are easier to work on than others. Others aren't worth working on at the moment, are being worked on by other, or are due for a rewrite. If you grab a ticket to start working on from a list in Trac (bugs.jqueryui.com) make sure it's in an Open status, that means it has already been triaged (someone verified it's a bug we'll accept a fix for) and is worth spending time on.
Recommended:(Widgets) Progressbar (no open bugs),
Button,
Accordion,
Autocomplete,
DialogThese are in good shape and generally a pleasure to work on, though some are challenging. Progressbar is very very simple (by design) so it might serve as a decent template for what a widget looks like and how it works. That (its simplicity) is also why there are 0 open bugs for it. Button is probably the next step up in terms of complexity. Then Autocomplete, Accordion, and finally Dialog. Autocomplete is the newest of the bunch, meaning it has an extremely tight and clean design and API. Its demos as well as some of
Scott's extensions show an ideal pattern for how we're now aiming to design all widgets, to be as minimal and extensible as possible. Accordion has some tricky bits often around animation (one of the biggest problems it solves). And Dialog, while very well maintained is not a trivial bit of code.
I can't promise any of these are simple (seems those are always the first to go ;) ), but if it were me, and I were brand new, I might want to start with this list (just grabbed at a glance at a few of the above reports):
ButtonAccordionAutocompleteDialogNot recommended:(Utilities) Mouse
(Interactions) Draggable, Droppable, Selectable, Resizable, Sortable
(Widgets) Datepicker
These plugins are very hard to maintain and are touched very little because of this. Much of the code is hard to understand and making changes is risky, so it isn't done unless very critical. All are scheduled for heavy refactoring (pretty much rewrites other than keeping some core API points) during the 1.9 and 2.0 cycles.
TestingOther than fixes bugs, another great way to get involved is running tests or writing tests (unit tests and visual tests). We're still working on good documentation on this, but let me know if that sounds interesting and I'll update the documentation.