In one scene during Lion-O's anointment trials where he's cornered by Slithe on top the mountain, Tygra knows he's not allowed to interfere but does so anyway and saves Lion-O's life.
At the end of part 5, Jaga tells Lion-O, even though he's now officially Lord of the Thundercats, he'll continue to see him should he need counsel.
Snarf going all badass to save some random Unicorn in the episode "The Terror of Hammerhand". Also counts as a Moment of Awesome for Snarf.
Both times that Lion-O was reunited with his father count.
Lion-O: Claudus, my father, wherever you are, my thoughts and love go to you. May we meet again, somewhere in time.
Any time Mumm-Ra shows compassion and love for Ma-Mutt. He even apologizes to him at the end of "Ma-Mutt's Confusion"!
Elaborating on the last point - Mumm-Ra has been verbally abusive to Ma-Mutt throughout the whole episode, even forcing him to help with the Weapon of Mass Destruction he's been using. When the Thundercats inevitably defeat him and blow up the weapon, how does Mumm-Ra react? Does he rant? Scream? Throw a tantrum? Nope - he frantically searches for Ma-Mutt, and upon finding him, gives him a genuine apology for being so cruel and promises to be nicer to him from now on.
Starts as a Tear Jerker. When Snarf (who feels he's becoming The Load to the now grown-up Lion-O) gets ready to return with his long lost nephew to his rediscovered home planet forever, the surprised but kind-hearted Lion-O doesn't give much protest before saying goodbye and that he will miss him - fighting back tears. It takes only a few moments from the ship's departure for it to land again and Snarf to jump back into Lion-O's arms.
Near the end of "Thundercubs", Maa-Mutt returns to the Black Pyramid and uses the cauldron to contact Mumm-Ra, who tells the dog that only the Ancient Spirits of Evil can rescue him from Thundera. They do so, and as Mumm-Ra emerges from the cauldron, an overjoyed Maa-Mutt jumps into his master's arms and is given a big hug.
In "The Mossland Monster", Snarf and his nephew Snarfer are part of a search party looking for a missing Tigra, when the duo are attacked by a 50 foot tall plant monster. In addition to its Super Strength, the monster also regenerates from all damage done to it and can fire beams of energy from its eyes that covers whatever they hit with thick moss, paralyzing its victims. Snarf has seen this thing effortlessly toss a spaceship around like a toy, but when it corners him and Snarfer, Snarf places himself between the monster and his nephew, and tries to reassure him that he'll do whatever he can to protect him. It's a very brave and touching moment for Snarf, who we see knows he's doomed, but is going to do whatever he can to protect his nephew anyway, even if its just making sure he gets shot first. Thankfully, Lion-O makes a last minute save.