I'll preface this by saying I'm a fan of ability checks. Always have been. I also understand and appreciate the ultra-light rules approach of OD&D. The more systems you bolt on, the more of a Pandora's Box you open up, risking making OD&D something it's not. Up till now most (okay, all) of the systems I've toyed with herein were modifications of existing systems (Chainmail for combat and Saving Throws, for example) rather than the addition of new ones.
However, I'm considering an Ability Check system.
"Why?" cry the traditionalists. "Why would you do something so...new school with OD&D!?"
A very valid question. And the simple answer is, because it works and doesn't hurt the game. It allows me to adjudicate pretty much any situation without resorting to judgment calls and DM fiat with which my players may vehemently disagree. So long as a system doesn't hurt or slow down the play, I'd argue, it's not necessarily bad to have a neutral way to accomplish these things.
This being said, it's also important to note that an Ability Check system should be used sparingly. If a human Fighting Man with Str 18/00 tries to hold a door shut against a couple Hobbits with strength 10 and 11, be serious: no roll is necessary. He can hold the door. But what if he's trying to hold the door against two other men with Strength 16 and 17, respectively? That's a more difficult task indeed. You could simply rule based on the story that he does it. You could make the other two each roll an Open Doors check. There's lots of ways to do it. An Ability Check system simply puts it on even ground across the board.
There are two ways to handle this in OD&D. The most basic is roll 3d6 and attempt to get under (not equal to or under) your Ability. Thus, if you have Str 17, you need to roll 16 or lower to succeed. Resisted checks are adjudicated based on whoever succeeds by a higher margin.
Another way is a "roll over" system with a target number. Since I've been using 2d6 for a lot of things, that seems a good option. The first step would be to grant Ability bonuses. The easiest way there is to import (and expand for % strength) those from the Moldvay Basic D&D books:
The average roll on 2d6 is 7. Therefore, the basic target number for a task of average difficulty would be 7. Bonuses and penalties for tasks of greater or lesser difficulty can be granted on the same scale as the table: difficult tasks are -1, very difficult -2, and extremely difficult -3, whereas easier tasks are at +1, simple tasks at +2, and basic tasks at +3, though basic tasks should rarely be rolled for unless there is some serious consequence for failure and the character is performing under duress.
Resisted tests between two people, naturally, function based on the higher roll winning.
Just some idle ponderings. At very least it functions as a fallback for situations where the DM isn't quite sure how to adjudicate.