I have been intrigued by the idea of removing attack rolls and just rolling damage ever since reading Into the Odd - but it was just too far a deviation from D&D overall for my liking. I did some math and I am going to try this house rule in my next session of B/X to see how it changes the game.
No Attack Roll House Rules! It's pretty simple. Remove the attack roll and replace it with a damage roll which corresponds to the character/monster's THAC0/BAB as noted on the chart below.
There are a couple strange ones at high levels (Hydras/Dragons) where the conversion isn't super clean and some monsters with powerful abilities (like turn-to-stone) do significantly less damage but for most of the manual it seemed to be pretty consistent. I actually think Hydras are fine. I think in OSE, which I was using as a frame of reference, they should have their HD doubled (they get 1 HD per head and WHOPPING 8 HP per HD). If you give them 2 HD per head (rolled for HP normally) it brings the creature more back into line with most of the other ones. This does take into account creatures with multiple attacks but not special damage attacks like dragon breath (I'd play those by the book since they largely hit automatically or require saves).
Armor acts as damage reduction instead of improving armor class. Now, my thought is that every attack does a min amount of damage (regardless of damage reduction) equal to the number of damage dice rolled. So 14th level Fighter (+9) rolls 3d6 damage always doing a minimum of 3, average of 10.5, and max of 18. I do not believe it would be game breaking to allow strength or magic to modify damage.
Edit: Wanted to add in here because I felt it was important. One of the goals of this rule was to only remove the attack roll without having to change any other aspects of the game (HD,HP,Stats,etc all stay the same) and have it play out to a similar outcome as the traditional mechanics.