If you are coming into a tabletop role-playing game, coming up with a character is one of the first things you'll need to do. And an important part of that is coming up with a back story for your character. However, it isn't an easy thing to do and can become harder as time goes on and your well of ideas starts to dry out. I know I still occasionally struggle with it but I hope some of what I do helps. If anyone has their own input, I'd be happy to hear it too.
Same Character Problem
It can be hard to play a new character sometimes. If you've played one for a while, it can be hard to switch to a new one. There is also another situation that can come up. If you are the only member of the party who died and your fellow players care a lot about party balance, they might very well want you to play a similar character in combat. This isn't a problem if you can switch from one class to another (wizard to sorcerer) but is one if they want you to play the exact same class. After all, that may be why everyone went with their classes: it created the best party. I won't be talking about the solution here, but felt this problem was common enough that it should be mentioned. Plenty of groups aren't nearly as prescriptive in their approach to forming a group. 5 bards? Sure. Why not.
Even if you end up making a fairly similar character mechanically, your choice of background can have a large impact on your character themselves. When I say background, I mean the big overarching details before the start of the campaign. It could be their place of origin, their profession or even related to their family. Such big changes tend to cause ripples through other aspects of a character so I find making this kind of change helps me come up with characters that feel different. In a system that lets you pick a background and have it result in a different character mechanically, the usefulness of this kind of thing is amplified because it gives you mechanics as well. I'd also include race in these section. It often comes with ramifications for who the character is in the world. If elves stay to themselves, why is this one radically different?
Even if you pick a different background for your character, they might effectively end up acting the exact same. The difference would just be their origin story. Some people I know tend to start off with a prototype of an existing character and change things to come up with their own. I occasionally use it too, though the background technique tends to work out pretty well for me. You can also lift elements from multiple characters into a brand new one. It can result in some characters that aren't too exciting because everyone has seen them before, but it can also result in non-typical characters as well. All it takes is using an obscure character as your inspiration.
One player I had wanted to make a spartan like character using heavy armour, spears and shields. Incidentally he had just seen 300 recently. People usually pick a weapon with a higher damage dice, but the player made it work through clever use of the thrown property and a backup weapon. However, he also wanted to have a character that would make sense in such a background and so they constantly fought against their desire for rigidity in the face of new circumstances.
Flaws vs Quirks
I tend to find that finding real, meaningful flaws for your character is an incredibly powerful tool. Since they tend to be meaningful and large, they tend to touch multiple parts of a character and force you to come up with a differences. However if you want a radically different character, I find it's important that they aren't quirks. Quirks can help make your character unique and memorable in a scene, but they often don't change the fundamental underlying character. Sometimes you don't want something radically different but different enough, and for that quirks can go a long way. I find that some of the best flaws involve thinking of the character themselves, and the problems they would face in such a world. Classic examples are naivety, prejudices, mistrust, greed, apathy, and discontent. The question once you have one becomes why? Why are they like this, or lacking that particular thing?
When you find one, though they can improve and get addressed over the course of the campaign, it is best that they don't all completely disappear. I've seen it before where 5 sessions in, almost all of the flaws evaporated completely. I've also seen where they became less severe, but the character had to agonize in not letting them best them. Having somewhere else to go or something else to address works wonders.
I like to give my players a heads up on what the world they will be playing in is like. Many of the Dungeon Masters I played with were the same. Players can even bounce ideas off the Dungeon Master once they know, and get some ideas to be better integrated into the world. It's collaborative story telling, and you shouldn't be afraid to talk about things with your Dungeon Master. Some elements can come up naturally later over the course of the campaign. Doing so makes the character far more tied to the campaign. It can also inspire people to come up with new characters after hearing about where it will take place. Naturally, this is greatly affected by how strong the setting is. That human factor can still often make it easier for people to get the ideas flowing.
As a Dungeon Master, don't be afraid to introduce some new mechanic things to make characters unique. It's can be tough to balance, but little things can go a long way. If your player wanted a character with divine origins, would it be that harmful for them to be able to teleport their weapon into their hand using a bonus action? Or set a minimum armour class if you planned to attack your fighter while they didn't have armour?
People in your group don't need to come up with their stories on their own. They can also team up to make stories that cross each other's path again and again. The great thing about this kind of approach is that it gives you someone to interact with and as a result come up with things you otherwise wouldn't on your own. Put another way, it becomes more like improv. This works great for some people can be very intimidating for others. One of the most memorable parties I had in the past involved to players who decided to play as members of the same organization. They were two fighters and a cleric of war on a campaign. Another involved two players who made their character brother and sister halflings.