Tess definitely occupies a little sister role in Day's life. Because he can no longer live a normal family life with his siblings and parents, Day takes on Tess even though he knows that she'll be a bit of a burden on his lifestyle. Perhaps it's because he misses his own siblings and wants to recreate some semblance of a family in his daily life, even if it will be a little more dangerous. Even so, after helping her for all these years, he comes to find that she's helpful—and she provides much-needed companionship:
When I first met Tess three years ago, she was a skinny ten-year-old orphan rummaging through trash bins in the Nima sector. She'd needed my help so much in those early years that I sometimes forget just how much I rely on her now. (1.5.34)
Over the years they come to rely on each other as a family. Tess has learned to keep up with Day and saves him whenever he gets into trouble—even when he breaks into the hospital and barely escapes and is direly injured, Day never doubts that Tess will do what she can to save him. She also wholeheartedly helps him raise money for his family, and never acts selfishly.
Tess will be there. She'll find me and help me to safety. (1.3.62)
She'd never abandon Day. They're family now.
Tess is also an extremely trusting and trustworthy person for a street beggar child. Instead of adopting the same caution and skepticism that other street children do, she instead chooses to trust people and care about them wholeheartedly. She obviously trusts Day immediately, and she also takes to June right off the bat, caring for her and dressing her wounds when she's injured in the Skiz fight.
Even after June betrays Day, Tess is willing to hear her out and is hopeful that she'll help them again:
Tess lets out a sigh. I exchange a look with her, and I can tell that this is something she's been trying in vain to convince Kaede to do ever since Day was arrested. (2.12.56)
Tess just wants to believe and trust in the people that she cares about—and she doesn't give up on people easily because of this.