It is a little weird to be talking about why I care about a character I have created, but it is a totally fair question to be asked- if I'm going to imagine that other people are going to be interested in his story, and find his struggles compelling, then there's no reason I can't account for that myself. So... here goes.
I care about Ian, one of the main protagonists in Smaller Totems, because despite the fact that he is a mess of a human being, he's still willing to fight, every day, to pull himself together enough to continue the search for his missing younger brother. Constant nightmares. Mental health issues. Bad organizational skills. A crap job. He just shoulders them all, as best he can, and continues the search. He's broken, but he doesn't let it kill his bravery.
Also, I care about Ian because he's willing to believe in dangerous things. When faced with a choice between seeing his dreams and intuitions as a sign of his failing mental health, or seeing them as a possible manifestation of a "gift" that might help him find his missing sibling... he forces himself to walk the tightrope between the two explanations for his experience. How do you not care about someone who is willing to go a little bit insane if it means there's a chance they can help someone they love?
But enough about Ian...
lets talk about the Ed, the robot from Tethered!
Ed manages to be endearing and caring, without leaning too heavily on the "naive robot" trope. It behaviorally runs the healer/medic gamut from nurturing caregiver to pointer-out-of-the-obvious, and manages to be very openly human in moments where Cara, the human he is linked to, seems incapable of doing so.
One of the things I enjoy most about Ed is that, name aside, there's an interesting mix of "masculine" and "feminine" behavioral traits, which keep me from being able to immediately think of Ed as a "he" or a "she". This is something I really appreciate from robots in writing. Some readers might not care about details like that, but to me their signs of a care and attention to detail and character that I really enjoy.
If you haven't read Tethered, you really ought to start!