She actually did once. However, we simply merged.
That's the idea, lucky you.
Coming back to Breezehome, I'll find her upstairs exclusively. Usually in the chair next to the door in my bedroom, but sometimes standing in her alcove, facing one way or another. When you enter a cell that's your home, the housecarl or other resident(s) will be randomly placed. I see this every time I step out of Honeyside only to go back in because I forgot something. Lydia never does anything but travel between those two points. Yes, sitting in the chair as soon as she quits it works. She doesn't ever cook, since she's never downstairs. I suspect this is due to a terrain issue with the top of the stairs. You may have noticed that you can't simply walk up the stairs and proceed onto the landing? There's a break there, probably enough to dissuade the pathing AI.
Now for a true story of Lydia on her own. We had gone to Honeyside and I decided to take Iona out for some fun and new gear, so I dismissed Lydia. On a whim, I decided to follow her as she walked the lonely miles back to Whiterun. Past Shor's Stone and down the hill, she killed a frostbite spider. Next she decided to take a shortcut down the hill where the road zig-zags. This must've added at least 10 minutes to her trek, as it was very steep, and she'd only take one or two steps before pausing for about 15 seconds. Nothing else of note happened until crossing the bridge approaching Honningbrew Meadery. Instead of logically continuing past and towards the stables, she took a left up the hill, as if going to Helgen. Somewhat past the meadery, she realized that was the wrong route, and timidly made her way down the rough terrain between the meadery and the farm. Once on the road again, she went home.
I mentioned in another post that Aela did exactly the same thing in the quest Battle for Whiterun, where I had to chase after her all the way from Windhelm. She also took some off-road excursions yet remained parallel to the road, occasionally getting confused and having to back-track. There must be some odd breaks in the terrain that mess up AI pathing as opposed to simply following your lead.