OD&D and BX thieves are... pretty bad. Even if you subscribe to the interpretation of their special abilities being quasi-magical, the chance of them actually working is still bafflingly low at low levels. This is my attempt to make the thief into a more playable class (especially at low and mid levels).
Back-stab: When attacking an unaware opponent from behind, a thief receives a +4 bonus to hit and doubles any damage dealt. At 5th level and above, damage is tripled instead of doubled.
Scroll Use: A thief of 7th level or higher can cast arcane spells from scrolls. There is a 10% chance of error: the spell does not function as expected and creates an unusual or deleterious effect.
Thieves have a few specialized skills they can attempt to use much like in BX. All of the usual skills (except climbing, which would get nerfed) have been consolidated into a single "Thievery" skill. The red number in brackets next to the Thievery score indicates the probability of something extra happening after using or failing to use certain skills (check the explanations for disarming traps and pickpocketing below).
Climb nearly sheer surfaces: Only roll for climbs that an adept real-world climber would find challenging. Roll a check for every 100’ climbed. On a failure, the thief falls at the halfway point and suffers fall damage.
Find and disarm treasure traps: A Thievery roll is required to find a trap and then another to remove it. The find traps roll can only be attempted once, but the disarm roll can be tried multiple times. Every time a thief fails a disarm roll, a subsequent d100 roll must be made to find out if the trap has been accidentally sprung. A trap is sprung if the result of this subsequent roll is equal or less than the red number in brackets next to the corresponding Thievery score. Each attempt to find or disarm a trap takes one turn.
Hear noise: In a quiet environment (e.g. not in combat), a thief may attempt to listen at a door or to hear the sounds of something (e.g. a wandering monster) approaching. Make a Thievery roll or roll under Wisdom, whichever is higher. Can only be attempted once per turn.
Hide in shadows: Requires the thief to be motionless—attacking or moving while hiding is not possible. While hiding in shadows, thieves use a special breathing technique which slows their heartbeat & lowers their body temperature and thus, if successful, are also undetectable by creatures with infravision.
Move silently: A thief may attempt to sneak past enemies unnoticed.
* The referee should roll for hide in shadows and move silently on the player’s behalf, if the roll fails, the referee knows that the thief has been noticed and should determine enemies’ actions appropriately.
Open locks: Requires thieves’ tools. A thief can try this skill multiple times per lock, each attempt taking up one turn.
Pick pockets: A thievery roll must be made to see if the pickpocketing attempt is successful, and then (regardless of the result) a subsequent d100 roll must be made to see if the victim has noticed the attempt. The attempted theft is noticed if the result of this subsequent roll is equal or less than the red number in brackets next to the corresponding Thievery score. If the victim is above 5th level, the thief’s roll is penalised by 5% for every level above 5th. This ability can also be used in shops and markets to steal suitable items.
I am of the opinion that non-thieves should be able to attempt most of the things that thieves can do, but their chance of success should be much lower. For most skills, I would consider giving non-thieves around a 10% chance and only one attempt (in the case of picking locks and disarming traps), and calling for a saving throw of some kind if there's a chance of springing a trap or getting caught stealing. Most importantly, always remember that the thief's abilities are specialized skills and that you should try to call for rolls or checks as seldom as possible. Pickpocketing or sneaking past a sleeping guard, for example, should be relatively easy for most adventurers, regardless of their class, and should at most require a dexterity check if the guard is an exceptionally light sleeper. Only call for Thievery rolls when a character attempts something that a competent thief would find challenging. Wrapping up, here are a few examples of situations where I would say a thief skill is being used: