When should an author scrap a character? When should a character be considered useless and without a purpose? Should all characters who don't play any role in the story be scrapped? Are these considered good enough reasons for a character to be part of the writings? - Being family member - Being a love interest ......... When they seem to have no other role than that throughout the story? When they only seem to be "filler characters"?
I suppose it mainly depends upon how concentrated your character population is anyway. Do they perform any function, however small, towardsdriving the plot? Will their non-existence cause any questions to arise (does she actually have a mother? Is he asexual?)? Can you enlarge their character to present aspects of the plot, e.g. a microcosm of the main issues, a different/contrasting perspective on religion/politics/ethics? Does the reader know they are unimportant?
Agree with all of what RowenaFW said. You could always use them as foil characters. There is always the option to use them add another depth to the story that your essential characters don't provide, similar to Fred and George in Harry Potter. eg. you could make the brother of your MC to be absolutely hilarious; add an element of humour in the book.
Sorry - welcome to my world of semantics. What do you mean by scrap? If you mean remove from the story, then... When they add nothing of value to your main story / plot. By and large, yes. Of course this relates to characters and not what I will call players. I think making an understanding of the difference between a character who has an integral part to play in the story and a player, who is simply a person in the world at large (in your story) is important. Not all players are wasted if they are part of the universe. If they have no role but you wish to keep them then perhaps they could be moved to a player - like in the movies - 'Third guy from the right' or 'woman by the water fountain' kind of thing. No. Not good reasons at all and most publishing houses, in fact, recommend against them.
All this is entirely up to you. If you feel a character needs to be scrapped - scrap it. If not - don't. What role they play is irrelevant, imo.
I'd say don't just scrap character's unless you feel like rewriting, but if said character can't add to the story at that point in time self em. You gotta realize that when writing essentially you're creating a world filled with interactions and millions of people. However if they are a semi-main character and you just don't want to deal with them anymore. (I know about having too many supporting characters) Kill them. Give a solid and reasonable explanation and then you're clear with closure.
If your character isn't pulling their weight in the story, something needs to change. As you have said, one option is to scrap them. As others have said, another option is to give them a bigger role. A third method that I've done before and liked the outcome is MERGE them with other characters. For example, I had two characters that were very similar. One was comedic and spunky, and the other was comedic and ditzy. They were each fine characters on their own, but they kept stepping on each other's toes. So, I merged them together and got one character that was comedic, spunky AND ditzy. She was more interesting, more developed, and now had more screen time. You can do something similar to your story. For example, are there any characters that could fill the role of being the love interest? Maybe someone would work if you changed their gender? A different way of thinking of it: could your love interest take over the role of someone else in the plot? I like this method because you don't really change too much in your story, you're just shifting things around a bit.
Here is a good way to decide if you can get rid of a character: Make a list of every character you have in your work. Next to their name, write down what their purpose/ goal is. If you are working with a giant sci/fi fantasy work, where there are several different lands or planets, and new characters in each land, you might want to do this exercise for each set of new characters in each land, or world, or whatever. Next to their goal, write down how close they are to the MC. Next to their relationship to the MC, write down a plus sign if their function is to help the MC to his goal. Write down a minus sign if his function is to help prevent the MC from getting to his goal. Now look at the chart, if there are any 2 characters with the same data, or almost the same data, than merge those characters.
What I mean with the word "scrap" is "get rid of". Did I use the word wrongly? I'm sorry if I did. I am not a native English speaker, but I try to use the proper word and grammar. Jowettc, how would you differ between a character and a player? Where goes the line?
However small a character's role might be it should help the plot development, character development... else, they become mare distractions. You shouldn't go on and on about a character just because he/ she happens to be the brother/sister of the main character. It sounds rather rude and selfish in real life but you have to ditch a character if you have no further use in your story.
Instead of thinking when I should scrap a character, I often consider if a particular character deserves to have a name. Using the OP's example, if a family member exists only to serve the plot, we can just name him "uncle", etc. I rarely complain about a character being unnecessary if she has no name.