One thing I did enjoy back in the day was rolling in order to figure out what I wanted to play. This led to some odd characters, and even ones with a random high ability score in places a character wouldn't normally have, like my 2e Fighter with 16 Charisma. I took the Cavalier kit, made him Lawful Good, and tried to follow the Paladin's code of conduct as best I could- his backstory was that he tried to join a knightly order of Paladins, but didn't make the grade.
Another character from the 2e era was a Dwarven Thief (Locksmith Kit) who had higher Wisdom than Dexterity, who I played as exceedingly cautious and who exclusively used a short bow rather than even attempt melee combat (making their backstab a completely useless ability, lol).
In both cases, it was pretty fun, even if not optimized. That having been said, I'm not advocating Int 12 Wizards or anything like that- I know a few "real role players" who insist that characters who aren't competent are better, but having watched them drag their parties down, I'm not inclined to agree. If you can meet a baseline of competence, and then have some unexpected strength, that can make for interesting characters.
Obviously, the same can hold true for an unexpected weakness, but ability scores have changed a lot over the years. In 2e, an 8 Wisdom wasn't a big deal. Now in 5e, it can lead to a character who gets mind controlled more often than Superman, which is a problem your party will not appreciate.