First, I'm assuming the type chart is always true.
Most mega stones don't change the holder's type. For example, a Steelixite Greedent remains normal type and has one weakness. I don't want to list all the Pokemon with one weakness and all the mega stones that keep their one weakness, but I can put this question back onto the unanswered list if you tell me this answer is incomplete.
The only allowed mega stones that change the holder's type are Aggronite, Altarianite, Ampharosite, Audinite, Charizardite X, Gyaradosite, Lopunnite, Mewtwonite X, Pinsirite, and Sceptilite, so the only possible type changes are -secondary, +fairy, +dragon, +dark, +fighting, and +flying.
I'm pretty sure it's impossible for a fairy, dragon, fighting, or flying Pokemon to have exactly one weakness, so all answers must use either -secondary or +dark.
-secondary (Aggronite): Hoothoot family, Pidove family, Diggersby, Stufful family, Kantonian Farfetch'd, Rufflet family, Oranguru, Drampa, Alolan Raichu, Toxel family, Helioptile family, Morpeko, Togedemaru, Rotom, Dracozolt, Arctozolt
+dark (Gyaradosite): Trubbish family, Croagunk family, Salandit family, Koffing, Kantonian Weezing, Skorupi, Mareanie family, Drifloon family, Duskull family, Gastly, Haunter, Pumpkaboo family, Galarian Corsola, Cursola, Litwick family, Mimikyu, Unova Yamask, Cofagrigus, Sinistea family, Phantump family, Dhelmise