I would like a widget that provides most of the functions currently found in the "Remote" app by Apple.
UPS and FedEx package tracking widgets would also be nice, so that if you have an important package coming in and you want to track it, you could access its current whereabouts and other relevant information in the Notificiation center.
Access, per se, to the app store is already free. Anyone can access it at any time. If you're suggesting free access to download apps from the app store, then...
My weather widget is "The Weather Channel" which does show the current temperature along with a brief summary ... "66F ... (location) Sunny with temperatures rising towards the low 70s. Winds NNE at 5 to 10 mph. Expect dry conditions over the next six hours" ending with an instruction to "tap foe more" that then opens the full app's 10 day summary.