There are two methods by which you can make the death of a relatively obsolete background character sad and meaningful. There is one thing you have to watch out for though:
This might just be me, but I never like cannon fodder - that is, a character who is created purely to die. They are usually indicative of an author taking shortcuts. For example, a writer might kill off some cannon fodder to show 'just how real' the threat is. That's a shortcut and sloppy style.
That being said, there are obviously times when you need to kill off a background character. When those times arise, they will always feel like canon fodder, unless you have taken steps to avoid it. And those steps are simple: introduce them long before they die. The further back you introduce them, the more they feel like part of the story, regardless of how much screen-time they have.
Comparatively, if you introduce them and then kill them in the next chapter, it comes across as you just creating someone to kill - cannon fodder.
With that out of the way, we can move on to the first of the two methods: making the character sympathetic. The underlying principle here is that you need to create a character the reader inherently doesn't want harmed, much less killed.
There are of course a host of ways to do that, but the best tend to focus on the character's traits - what make her who she is. As a background character, you don't need to put too much thought into this; as long as the traits make her enjoyable versus mean or annoying, you're on the right track. Kindness, optimism, loyalty, even things like having a sense of humor will work.
Since the character doesn't have much screen-time, that's all we know about her, and with no other information she becomes that trait. Thus when you kill her off, it's like killing the trait itself.
If you really want to get your reader (and your characters) in the gut, go for an injustice. An injustice is a concept I've recently discovered and been experimenting with, but it comes down to a simple formula: an innocent is caused intentional harm.
I use injustices as drives - that is, what motivates the character to complete the goal - because they have the potential to be really powerful. But you can easily use them to lend a death a lot of emotion.
The first thing is to make sure the character dying is an innocent. That is, someone who is - or at least is assumed to be - innocent in the matter at hand. Think of the 'innocent bystander'. That's the kind of person you are going for. Children tend to be seen as more innocent than adults, and the defenseless as more innocent than the capable. I'm not saying whether they are or not - it's just human nature to see them that way.
The next step is to make the death intentional from an outside force. Someone killed the innocent, either directly, or indirectly (ie, forcing them to suicide still works).
Look for the motives. History is full of injustices. Just do a search for 'worst moments in history' and you'll find plenty of examples, among them things like the Holocaust. The motive is important. The death can't be accidental, even if the harm wasn't aimed personally at the innocent (again, think of the innocent bystander). Think of the most corrupt self-serving motive which works for your story, and make that the reason the innocent died.
Injustices generate a lot of emotion because they can be boiled down to something inherently wrong. As long as that inherently wrong thing is largely accepted by the world as wrong, the reader will want to fix it. Anyone would. The reader therefore doesn't want the background character to die, and feels sad at the death.
So for example, think of the Holocaust. Innocent Jews held in concentration camps, where they are starved, overworked, and eventually murdered. That boils down to a lot of things that are considered wrong by basically everyone: The taking away of freedom, unprovoked cruelty to someone who can't fight back, wrongful accusations due to religion... there are a lot more, but those are just some examples. The majority of people would agree that these are things which cannot be tolerated.
NOTE: The nature of an injustice goes beyond sadness. The reader (and the characters) will want to fix the cause of the injustice. This is natural, so make sure it works for your story. An injustice can easily 'out-emotion' any drive you have, unless they are the same thing.
Hopefully this helps you. Pick the method which best suits your needs. Let me know if you need more information.