Six ways...
If I were designing a familiar from the ground up...
The familiar is more like a teacher or mentor than a sidekick. You keep them around because they know more about magic than you do (for the brash young spellcaster) or because they can assist you in magic rituals and arcane trivia (For the more seasoned archmage)
That's not a role that would be easy to fit into D&D though.
My first way would be to focus it on this. Familiars teach you spells. The spells you learn from a cat are different from the spells you learn from a toad, which are different then the spells you learn from a weasel.
Number two would be to focus on their otherworldly nature. Make them part of another plane by default, the Feywild or the Shadowfell or Somewhere Else. They gain magical abilities related to thier particular planar connection as they level up.
Number Three would be to make them decent "sidekicks." They actively use their actions to do things like Aid Another, they carry useful little abilities, they can be something the spellcaster can "ride" when they're in a bad scenario. Have them serve as good Allies, making what the spellcaster can do on their own even BETTER with the familiar around.
Number Four would be to make them free. You don't loose XP if they die. You don't spend gold to get one. They're just *there*. Though if you loose it, perhaps you can't learn as many new spells on your own.
Number Five would be to make them active NPC's in the character's story. They are sagacious otherworldly beings, let them use their knowledge and their connections to help the PC's get there. Familiars are guides on other planes, diplomats in front of delicate NPC's, they know other wizards, they serve as the spellcaster's connection to their college, etc.
Number Six is the old "true soul" myth-trick, where the familiar reflects elements of the wizard's own personality cast in animal form, so that as the wizard grows and changes, the familiar does as well.
Think of the Daemons from
The Golden Compass, and combine that internal externalities with an otherworldly nature that lets them be at once something less than and more than mortal.
Driddle said:
I don't really think it needs to be argued. It's common enough to simply forget you even have a familiar (as OotS got comedy out of). If your experience differs, I think it's safe to say your experience is the anomaly.