Here's what I envision:
Start with the PSP Go form factor. Add touch capabilities to the screen, as well as a second analog nub, tilt controls, and front and rear cameras.
No UMD drive, so PSP discs would not work. However, the unit would be compatible with downloadable games for the original PSP.
Physical games could still be purchased at retail, as a selection of games would be produced on ROM versions of Memory Stick.
PSP2 could connect to a TV, just like the more recent PSP models. PSP2 games could be played at 720p resolution when connected to an HDTV.
PSP2 would have more advanced access to PlayStation Network, similar to what is available now on PS3, with full friends list support, messaging, voice/video chat, and trophies. PSN data could also be communicated to popular social networking sites as well.
A much more advanced Web browser than what we've got with PSP now.
Other network applications: Netflix, Hulu, YouTube, Pandora Radio, Skype (with video), Google Voice. Existing Internet radio function would also be spruced up. Games could export video clips to YouTube (like you can do now on PS3), and Internet radio or Pandora could be used in the background during a game like a custom soundtrack.