This has been asked for before, so you're not alone.

As far as I know, the only way to do it now is to delete the files manually from the Documents\My Games\LegendaryHeroes\Race and/or Documents\My Games\LegendaryHeroes\Units folder.

Make sure you have Steam cloud sync turned off, though, or they might come back.

Does deleting a custom faction via replacement work for cloud sync as well?

You can delete a custom sovereign  by right clicking on it, and then clicking yes.

I don't believe you can edit a faction, unless you delete, and add it again (with the new changes) manually. 

Creating a custom faction with the same name as one you already created does effectively edit the pre-existing one. I think the person talking about right clicking is mistaking custom sovereigns for factions.

And

Does deleting a custom faction via replacement work for cloud sync as well?

 

it apparently does.