Orihime Inoue
little sister from another dimension
It's hard not to like Orihime. I think so, and seems like the Bleach folks also agree. Rangiku treats Orihime like her own little sister, partially because she's simply adorable, partially because there's no girl like Orihime in Soul Society: she's just so clumsy and unstable that you feel like protecting her regardless of what she might say. The latter cannot be helped in a sense: neither had Orihime been trained like the Soul Reapers have, nor is she used to grand scenes of life and death. Rangiku understands that, and as a result, appears extremely mellow on her.
When Rangiku first arrived in the human world (before half-forcing her way into Orihime's room asking for a place to stay), she immediately noticed the dark clouds surrounding Orihime and the other kids. Not only that, a woman's instinct told Rangiku that Orihime was carrying a different type of problem than the rest: it was something more personal. Before long, Orihime poured all her worries to Rangiku, who was all for ears and reassured the young girl with kind words plus passionate glomps ;)
Naturally, such is what a big sister would do. But Rangiku was also trying to remind herself that everything must have a reason to happen, more or less in reference to her own situation at the time. Rangiku was probably aware that deep down, she would like to speak her real thoughts just like Orihime did in spite of whatever consequence that might accompany in the long-run. She knew it's her duty to fight, but it's no easy task having to face the man who had been your only family over the last couple hundred of years. In that way Orihime is like a mirror to Rangiku's carefully hidden feelings, and Rangiku must have found some comfort cheering the girl up: she had to keep her own inner world from collapsing, to reiterate what she believes is right for the benefit of everyone.