Subvert people's expectations - If the character is a child, show her to be strong and independent and wise beyond her years, but only because you know she's used to her parents not being around and doesn't know how to ask people for help. Show the player that an ogre can be kind, a vampire persecuted by hunters only wants to be accepted. By surprising players, you convince them that this character - and the depth you're hinting at - is worth saving.
Humour, personality and silliness - This can be applied in so many ways. We all love to laugh, we all want to be entertained. Who doesn't want to rescue a lovable Purple Hippo terrorized by an equally silly villain? Less extreme, the rescuee can also win players over with their jokes and/or witty sense of humour. There's a reason people will usually always remember the characters that make them smile, be it a movie, book or video game. Simply put, the more you convince players that this character would be a fun person to have around, the more they want to save him/her.
(You can also use humour to apply through the lenses of your main character. Suppose, for instance, that your main character is a goblin who wants to be a paladin. He's convinced that saving people is the only way to fulfill his lifelong dream of getting into the Holy Order of Stuffypants. The more appealing you make a main character with the motivation to rescue people, the more players will get into the 'role' of goblin paladin wannabe and naturally want to fulfill that goal. But that isn't really answering your question, I know).
Building bonds and relationship - If the character you're supposed to save, someone your main character has never met before? Or someone you already took the pains to carefully build up a relationship around? I'm not talking about romantic relationships, I'm talking about using up game hours to show (not tell) the player just why this character means or should mean a lot to your main character.
I'm not explaining this very well (a little sleepy) so here's an example. Say your main character is a bitter war veteran and she's living out her retirement years in the forest, alone. Along comes a scarred elf who teaches her that humans and elves can get along after all. He's wounded, and she helps fix his leg, sets him up in a shack nearby. Her cabin is wrecked in a storm so together they both fix it. Winter comes and they work together to gather food and firewood to somehow survive the wolves and the cold. During dinner, they talk about growing crops instead of hunting all the time. She's reluctant, but the elf cheerfully persuades her to try planting vegetables. For the first time in a long while, she's hopeful and looking forward to the future. And then soldiers come crashing in one day, accuses the elf of treason against the king, knocks her out and drags him away. When she wakes up, the elf, her closest friend in years, is gone. And of course she frickin wants him back. In game terms all this can be shown with some cutscenes, maybe a tutorial of the two fighting off winter wolves, before the real game begins.
Really long example, I know, but the trick I like to do is make the players fall in love with a character. Make them sad, make them happy, connect them with an emotion they might have in common (such as the need for friends, or family bonds, or both wanting to be understood by an uncaring world.) Bring the player and that character to that perfect moment of understanding. Then, rip out their heart and twist as you take away that one thing precious to your main character. If you did your job well, your player would be screaming to knock down doors and kick butts to get that person back.
Kids and animals - I don't mean to lump them together, but it's true. Unless you have a heart of stone, it's very hard to ignore an animal or child in distress. It's why the video game Last of Us was so effective, why it made you want to protect a character who is (seemingly) helpless and dependent on you for love, warmth, support, safety. On some level, you don't want to let these kind of characters down, especially when developers know how to tug on the heartstrings.
Strength and usefulness of the character - Cynical as it sounds, sometimes us players just want to save a character we know would strengthen our party. We want them for the tank and meat fodder potential. We want them for the spells and the cold hard numbers that would allow you to beat that previously overpowered boss, thus earning you juicy XP and spell upgrades. Lord knows more than once, I wasn't really paying attention to Cleric Timmy's tale of woe of whatever monster kidnapped his girlfriend. I was rubbing my hands and thinking to myself, "Heal Alls at Level 5."
Edited: Family members - If the family member of the main character is in danger, be it a son, brother, wife, on some instinctual level most players would want to get them back too. It's built into us as humans - you might say it's a survival instinct, an evolutionary innovation. It's a powerful bond, which is why they say blood is thicker than water.
