Admit it, you smiled when Eli gave Rush his glasses.
Chloe's interactions with her father in "Pain". She is entirely aware that he's just a hallucination. She simply doesn't care. It's especially heartbreaking when Eli and Scott literally have to drag her away from the vision to force her to get cured.
Goes Tear Jerkery in the end of season 1, with Eli and Chloe discussing their friendship.
Pathogen, when Eli's depressed mom gets on Destiny via the communication stones, gets to see his face for the first time in months and tells him how proud she is of him.
Greer and Chloe's quiet moment in Visitation was strangely heartwarming they were talking about how to execute her.
A point can be made for Rush's speech to Eli in "Resurgence" being this - it's rare to see Jerk Ass Rush respecting someone.
In "Hope", Greer almost immediately volunteers to donate his kidney to save Volker, as well as his constant assurance that everything will go perfectly fine to everyone else involved definitely counts.
The real moment is just before this. Volker tells Greer about how he used to watch the starts in his backyard. What does Greer do? Takes Volker to the domed hydroponics lab, the best place on the ship to watch the stars. That, more than any other moment, shows that Greer really cares for the crew.
In Epilogue seeing the development of Novus from a small settlement to a prosperous metropolis fit the bill. For all of their differences, their squabbles and the adversity that they faced, the crew of Destiny were able to band together and create a future for themselves and their descendants.
Also, Eli Wallace apparently does not care that a problem has stymied Earth's brightest minds for decades. He's going to cure Lou Gehrig's Disease. Why? Because T.J. has it.
The alternate timeline where they tried to dial in a star. Rush makes a short speech about the importance of Destiny's mission, and asks for volunteers. Young has already agreed to stay with him, and nearly half the crew volunteer as well. It actually brings Rush to tears. Subverted moments later when Telford starts yelling how they're just going to get themselves killed.
Telford: I have half a mind to force you all back.
Greer: I'd like to see you try.
Young's speech about family in the final episode, "Gauntlet", as the final group of the crew - our main characters - prepare to go into the stasis pods for three years, or possibly forever. He refers to Scott as a son, Chloe as a daughter, Wray as a sister, TJ as the most compassionate, Greer as the bravest, Eli as the smartest... and Rush as the crazy Uncle "who, despite everything, manages to come through for you in the end."
The climax of "Light". Having spent an episode resigned to their fate, the elation of the crew as they realize that they're going to live is truly amazing. For the first time since they arrived, everyone is smiling.
"Sabotage". The simple act of returning Rush's lost glasses.
The appropriately named episode "Hope" is just covered in these, the most prominent being Greer saying he'll give up a kidney to save Volker, who's dieing of renal failure without even giving it a second thought.
"Cloverdale". Seeing the entire crew through Scott's eyes, set in Anywhere, U.S.A., just before getting married to Chloe. Having a bachelor party with his best friend, Greer, his father, Young, and his future brother-in-law, Eli. Even getting a lift home from the sheriff and his dad's best friend, Telford.