For example, at Alex's table, we use a death & dismemberment table when reaching 0 HP rather than just instadying, and there are also some extra rules regarding shields and helmets.
I think each GM or group probably house-rules the bits of the game that stick out the most, and in my case, a strong candidate for this is weapon damage.
There are two systems you can use for weapon damage: in the simpler one, all weapons, dagger to longsword do 1d6 damage. In the more complex one, bigger weapons do more damage. Neither is exactly satisfying. In the former, why bother carrying anything bigger than a dagger? It's just expensive and heavy. In the latter, why carry anything but the biggest sword you can afford? (And as it happens, the cost for even the biggest sword is a fraction of what gold you need to spend on ale and whores to reach level two, so you'll have that big sword pretty quickly.)
My alternative suggestion is to introduce Reach as a concept. Weapons all do 1d6 damage but have different Reach: your bare hands have zero, a dagger one, a short sword or hand axe two, all the way up to a polearm, which has perhaps five. Going up against someone with more reach than you means that you get the difference subtracted from your rolls and damage, and they get it added. For example, a dagger against a short sword in this system would be like a dagger - 1 versus a short sword + 1.
In that case, why ever use anything short?
First, different rules apply in close-quarters fighting. Reach is ignored, but weapons with a reach greater than 2 gain a penalty as big as the extra reach. Use a polearm in close quarters, and it's like a polearm - 3.
Second, there is a special rule for daggers: they grant an automatic surprise attack at close quarters, and you can carry a dagger without it being apparent that you're armed. Basically, they're for cutting people down in taverns and dark alleyways, not for proper fights.
I haven't played with these house rules, but they could potentially be fun. Players might start carrying a short sword and a spear, like roman soldiers did, or settle on a longsword as a reasonable all-round weapon. Others might just stick with a knife.
Suggested reach values for Labyrinth Lord melee weapons:
Of course, you could further complexify this by adding variable damage, at which point you might be spending too much time on dice for an old-school-ish game - but the combination of price, reach and damage would probably let you situate nearly all weapons at their own particular points of balance, offering trade-offs of damage versus reach versus suitability in close quarters versus price.
Finally, I personally think that the cost of weapons in LL is rather low: doubling weapon costs, and perhaps offering +1 weapons at very high prices, would keep the game of "what kind of weapon can I afford" interesting for a lot longer.