Jeremy Zawodny asks, "What web services do you wish Yahoo offered?"
I'd start with the basics. Web services for search and browse for the web, images, and shopping. Improve the visibility and expand the functionality of the existing news web services. Access to Yahoo Finance data (stock quotes, etc.) would be great too.
Google only provides access to web search through their API. Amazon only provides access to shopping. If Yahoo just provided a comprehensive suite of the basics -- web search, shopping, news, images, etc. -- they'd already be the strongest offering out there.
Update: Only a couple weeks after I wrote this, Amazon expanded their web services and released an Alexa API that gives access to web search, directory, and other interesting data. Yahoo still has an opportunity, but things are moving fast.