Opera would be easiest.
If you are commited to making a flash file to a homebrew app it can be done, but would take an enourmous ammount of work unless you were familiar with both stripping the images from the flash and coding a new backend for it in dev, so you would need to be a very good coder.
I would love to see something that allowed flash files to be converted to some sort of .dol, as there are soooo many great flash games out there that imo are WiiWare quality, and free.
If this all sounds horrible and you want an easy way, setup apache server on your home pc. Make a page that you can save in your Wii's internet channel as a favourite. Now if you want to make it "look" like a game do this:
Make a page that looks like what you want the channel to be, so a big picture say. Navigate to this picture and save it as a favourite on the Wii (it caches a thumbnail
*dont delet your net save file* ), it will now look cool. Next replace that page on your pc with the flash, and name it the same, so easier if the pic, and flash are both put inside an html, or php page. Address doesn't change then.
If your rebuilding or changing the .swf, just place it back in the html/php page and it will sync well on your wii.
I'm not gonna explain anymore, I'm sure you get it. I play with flash a bit, and have a similar setup in my favourites. One of the "channels" favourites is called arcade, I drop .swf files in a directory on my pc, and they are auto listed as links on the left of a frame page, loading in the main window. That's why I have apache, to save me typing out lists and making them links. If you need help with code, give me a pm.
Anyway, I know this is not as .wads on the main Wii menu, but it looks pretty close if you get it right.
To make it .dol = months of work and not a 1:1 copy, but re-written.