My first thought would be to use bamboo which is a hardwood but mostly hollow so it would also be light as well. Plenty of plans out there that involve hybrid bike frames with bamboo. Like this one from instructables.com from BambooBiker
You don't have to pigeon-hole yourself with hybrids like what is pictured above. You could easily make the entire frame from bamboo.
I have not built one myself but did look into it a couple of years ago. The great thing about the bamboo is you don't need to do a lot of shaping with the wood since it is already cylindrical in shape and lends well to modern bike designs. Note that there are many varieties of bamboo and they are all not as strong as the next. I have some bamboo that grows in Canada but it is not near the strength of the typical bamboo you would imagine.
I would consider laminated plywood as well for its dimensional stability but the might jack up the weight a bit depending. So many wonderful pictures of these on the internet. You can see some with, what I will presume is, steam bent wood so the frame is one visual piece. There are some open frame plywood bikes as well (hollow inside as supposed to being a solid frame) so that would be a consideration as well. Between this and bamboo you would have an easier time acquiring plywood.
Oh be careful for cheap plywood. Some like baltic birch comes highly recommended.
As for other woods I don't have any specific advice and more of a warning. Having a lighter weight wood could have a trade off of stability. Lightweight and durable don't always go together. Shock resistance would be a key feature as well. Some of the woods that do fit those criteria might be too expensive for use for a bike. This off course comes back to availability again.