I propose that magic items with static bonuses, like a +1 sword, should instead scale with level, increasing in power as a PC does. I am surprised that this is not more common, to the point where it should really be the norm instead of a house rule.
The first advantage to this is that it reduces the number of magic items one needs to include in the game. Instead of needing to figure out how two +3 swords, one +3 short sword, one +3 bow and one +3 mace are going to show up in such a way that the PCs can get ahold of them in a timely fashion, one can have them find two magic swords, one short sword, one bow and one mace along their adventures. As they grow in power and accomplish heroic deeds, they can be assured that their weapons will develop in power along with them.
The second advantage is that it rationalises the incentive for more powerful magical items. Rather than trying to find treasure and bumping into adventure along the way, characters are incentivised to go out seeking adventure, to perform great deeds and accumulate glory in order to increase the power of their items. It doesn't force characters to get into silly scraps with wandering goblins, or push them to fight every single opponent they find, but rather encourages them to perform truly heroic feats.
There are two possible frameworks here that I think would be manageable in play. I leave it to you to choose which you prefer.
Framework 1: Item XP
Items have levels. They provide a bonus equal to their level divided by 4, round up. The level of an item is based off the owner's XP total, but using the wizard XP progression. i.e. a character with 2500 XP and a magic sword gets a +1 bonus, even if they are a thief and are therefore 3rd level at 2500XP.
Framework 2: PC XP
Items have levels. They provide a bonus equal to their level divided by 4, round up. The level of the item is based on the PC's level, so a 3rd level PC's items count as 3rd level. This means that clerics, thieves, and other rapidly progressing classes will always have the best items.
Framework 1 is good if one wants rough parity amongst item levels for all PCs, while framework 2 provides extra incentives to playing thieves, clerics and other rapidly progressing classes. While I initially preferred framework 2, I have come around to preferring framework 2.
These are by no means the only possible ways of doing this, but simply two I think are simple and quick enough not to bog down play. I first encountered this concept in Midnight, the third-party supplement for D&D 3.x that depicted a low magic world where there are simply not that many magical swords for PCs to get in the first place.
One of the developments of that concept that Midnight suggested was unlocking powers. As a PC reached 5th, 10th, 15th and 20th level, the item would bond with them and unlock new abilities, or improve on old ones. I think this is a good idea as well, though this will require some foresight and planning on the part of the DM. Here are a set of general schemes for DMs to help them go through weapons and armour power design. These are merely suggestions, and I haven't playtested them extensively like I have with the upgrading bonuses.
5th level: +2 to hit and damage against a specific kind of foe
10th: Deals an additional die of damage, but of a different type (fire, magical energy, lightning, etc.)
15th: 1/day utility power equal to a spell of 4th level or lower (Knock, set things on fire with a touch, Dimension Door)
20th: The ability to incapacitate a foe who fails a saving throw in a single hit, either by paralysing, crippling or killing them
5th: Resistance to a single kind of special attack like petrification, fire, death attacks, etc. (saving throw for no damage or effect)
10th: 1/day utility power equal to a spell of 4th level or lower (Fly, Invisibility, Protection from Normal Missiles)
15th: Resistance to a second kind of special attack
20th: Immunity to the two kinds of attacks the bearer previously had resistance to, or the ability to reroll all saving throws