My favorite low stat character of all time?
Dingus! My Gully Dwarf fighter. I don't even remember the specifics- this was mid 1990s, and our games usually included plenty of funny smelling "green tobacco"... I just remember that he had very low scores in both intelligence and wisdom, not so great Charisma and Dex, and average strength. He wasn't even my creation! I owned the Dragonlance 1-4 module compilation and the campaign setting and such, and was reading through them one day when I came across a very evocative picture of a gully dwarf sentry. It was a skinny bearded guy with a helmet 2x too big, clutching a spear- and peering suspiciously with a one eye squinted sideways look. I showed the picture to my DM, and said "I want to meet THIS guy!!"
And we did, a couple of sessions later. We talked our way past him, and ended up impressing him so much, he gave us a guided tour of the gully dwarf city- and introduced us to "Da boss!" When the party moved on, he followed us. (Dingus hero worshipped our bard, and was in awe of our Elven Wizard) The DM then handed me the character sheet, and said he's yours now! Dingus provided quite a bit of rp fun, and a lot of comic relief.
Of my current 5th ed characters, Gary the mage has by far the worst ability scores. 9, 10, 11, 15, 14, 8, and I rerolled a few quadruple snake eyes... And he is human, so he got a +1 to every score, lol. I'm not even sure WHAT to do with my Ability score increases... Shore up those with painful penalties- like that 8 charisma? (thank gods we haven't run into any spellcasters that like spells with cha saves!) Boost his AC and remove that str penalty by giving each a +1? Maybe Con, for more health and better saves? Oddly enough he doesn't get hit very often- and usually makes his saves. The abjuration school has been a good choice so far.
I play him as a mid level noble's 4th son. Technically a noble- but not in line for anything. Thus he ended up a Wizard's apprentice and researcher. He's smart, perceptive, and has good horse-sense- but rather under-developed people skills. Not much of a talker, and little patience for fools or overly glib types. He IS, however VERY loyal, and those who impress him and win his trust can count on him. He's lawful good, a follower of Azuth, and was raised in a family that revered Torm. Gary is of Chondathan descent. He, and Paddy- our Cleric of Lathander work well together, and constitute a real 1-2 punch.
His career as an adventurer began when the wizard he was apprenticed to decided he'd taught him all he could for now, and it was time for Gary to put his lessons to practice and get some real world experience. So he sent him with a caravan, and letter to deliver to a colleague in Silverymoon. Bob and Frank happened to be along as guards. Paddy was encountered along the way, as a fellow traveller. Gary fairly quickly became fond of Bob and Frank- and their easy banter and lively senses of humor. They reminded him of his older brothers and cousins. He admires Paddy for his open hearted nature, generosity, and his charismatic personality.