I've been trying to figure out a way to allow my artificer to have a homebrewed quarterstaff that can also be infused with the "enhanced weapons" infusion. But for whatever reason, any and all homebrewed items are unable to be infused as they don't show up as an option on the infusions list. Is there a way around that? 

The Homebrew system only lets you create magical items, which cannot by infused by the Artificer.

Infusing an Item

Whenever you finish a long rest, you can touch a non-magical object and imbue it with one of your artificer infusions...

You will have to infuse a regular quarterstaff, or use your homebrewed staff and find something else non-magical to infuse instead.