So, like many others I had Lydia as my first companion. She saw me through alot, and I grew fond of her, despite the passive-aggressive "I am sworn to carry you burdens" delivery. (She looks sexy in an elven helmet, BTW). But, as I leveled up further, Lydia would die more often. It grew irritating having to reload gamesaves just to keep her alive. It got to the point where the wonky AI would let Lydia get stuck in odd places and succumb to traps, and I'd have to have her wait just to get past certain areas without her dying. This is no way to adventure!
Then I learned that your companions don't really level up with you, and their stats are mostly fixed in place from the moment you recruit them. Ahhhh....so Lydia couldn't really hang anymore.
If you're fond of Lydia, but have outgrown her as an adventurer, I recommend buying the mansion in Solitude. It costs 25,000 gold (if you don't use the exploit), so you're not likely to buy it until you've put in the time and are fairly powerful. In addition to weapons rack, enchanting table, etc., your thaneship comes with...Jordis the Sword-Maiden!
Jordis is like Lydia 2.0 She's just as attractive (with blond hair), and isn't whiny about carrying your dragon bones / scales. She appreciates healing-hands magic. She's more happy to be there than Lydia ever was. And she's likely to be MUCH more powerful. Give her some decent weapons, deck her out in good armor and have fun. Lydia can chill in Breezehome, keeping the fire going.
Just sayin....
Jordis is always a B*tch to me :D Whenever I go into the house, she's always saying "lost, friend?" or "you don't belong here!" followed by "how can I serve you, my thane?" If she actually forgets what I look like every time I leave, I think she has short term memory loss XD But I would note that while she is quite similar to Lydia, she has one of the two generic Nord female voices, unlike Lydia's which is semi-unique. That's pretty much enough to count her out for me, I end up going for someone more badass like that Khajiit, Kharjo, or Mjoll the Lioness.
Well, it turns out last night that Jordis bit the dust! During the Mage's Guild Questline, while defending Winterhold against those Magic Anomalies. After the fighting was said and done, I looked around for her and she was....on the ground dead.
I decided not to reload a previous save, and I've decided that from now on I won't reload a previous save if a companion dies. I have Lydia back in Breezehome safe and sound, and that's it. My followers know the risks when they choose to run with the Dragonborn! And as I'm learning, there are lots of available followers in Skyrim, so it's best not to get hung up on any one particular NPC.
So for now, I'm running with Mjoll the Lioness.