It's possibly the type of WiFi signal you're trying to connect too. I've been refurbishing some busted PSPs for a while now and when I fix them up at home and try to test the WiFi connection, my router signal isn't supported. When I bring it to work, I can connect there and update or whatever else I need to do with it.
If you're still having issues with it then your WiFi PCB may have been damaged some how. With that, you'll want to purchase a new one (shouldn't be too expensive on eBay) and follow this guide to remove most of the PCBs and get at the WiFi board: PSP 1000 Teardown