I wish you and your family well in 2021, troll
Taking the OP and this bit at full face value …
You’re right that
We should take care of her. Not strangers on the internet. But I stress “WE” as in my brother and sister and I
But the magic word there is “should”. Which is a normative statement about what ought to happen, not a factual statement about what has happened, is happening, or will happen.
Your brother and sister have, over a period of years, demonstrated their character to you and to your Mom. They are thieves and conmen at worst, and cynical selfish advantage-takers at best.
Your Mom has also demonstrated her character. Passive and victim-y. As she becomes elderly, she will naturally become more so; that’s built into human nature.
You face a choice. Pay for Mom 100% solo with zero help from bro & sis. Or don’t. If you do pay, accept that 100% for certain some of whatever support you give Mom will be diverted to support your bro and sis instead.
Whichever choice you make, own it and all the consequences for all involved. There are no good answers here.
Ranting / venting can be fun, but if it’s an excuse for indecision and inaction it’s unhelpful to your cause. However you understand it.
I’m also getting a sock vibe off it. I assume it’s a previously banned troll.
Absolutely. The coup thread alone is pretty much proof. You join and immediately start posting like crazy, including stuff that is deliberately inflammatory and disingenuous? Oh yeah, sock troll for sure.
Well, there’s your problem…
That’s my one free bite at this apple. 
We shall see. I explicitly offer no opinion one way or the other.
Don’t feel bad - the poster didn’t trigger my sock radar warning either.
I have no idea what the backstory is to this (I mean the hostility and suspicion directed at the OP) or if there’s anything to it.
Taking the post and poser at face value, here are a couple of suggestions for the OP:
First, start another thread on this outside the Pit. You might get some helpful answers.
Second, you and your mother (assuming she’s competent, and you don’t indicate that she’s not) should sit down with a lawyer. There are lawyers who specialize in what’s called “elder law.” You should find one in your state, with considerable experience.
It’s hard to find a good lawyer in this situation. Many lawyers view the elderly as a source of available money. Kind of like a junkie views the all-night liquor store. (You can find my pit threads on the subject of lawyers and my elderly and incompetent father).
But find one. Lay out everything for that lawyer. Find out what options are available to you and your mother. And have the lawyer work with you to protect whatever assets she has, and to recover anything that can be recovered from your brother and sister.
And I reiterate, start a thread in another forum. IMHO is probably best. GQ, maybe.
The OP reads like it was created out of whole cloth by a Topper – You only lost $2500? Well, my mom lost $100k!
Why should brother pay mom back? She invested in his business, it failed. That happens, and investors lose money all the time.
Also, is sister paying rent to live in mom’s house?
There’s a difference between lending someone money to start a business, and investing in that business.
If you, for example, take out a second mortgage on your house to start a business, whether the business succeeds or fails is irrelevant to your obligation to repay the loan.
But if your friend invests in your business, and has equity in the business, he/she is sharing the risk of failure. If the business fails, you both lose. If the business succeeds, you both profit.
Follow-up:
It’s the difference between a publicly traded company issuing stock, and selling bonds.
If a business fails (i.e. ceases to exist) both stock holders and bond holders are out their money. Informal, “here’s some money to help start your business” I should think would expect the same, unless there was some other understanding at the time.
We can debate this analogy endlessly. Suffice it to say that stockholders and bondholders are different. They’re not the same thing.
Perhaps my analogy to a home mortgage as a source of funds to start a business is better.
In any event, we don’t know what the arrangement between the OP’s brother and his mother was.
I still say that they really need to sit down with a good lawyer and figure this all out.
This is the key.
Good lawyers cost money that fictional people are unlikely to have.
I am the primary caretaker for an 88-year-old man with dementia. He is my father.
My relationship with some (not all) family members during this has been, to say the least, contentious. There is one sibling who is, shall we say, very interested in preserving my father’s estate. As opposed to me, who is only interested in deploying whatever assets my father has to maximize the quality of care he receives and his comfort and happiness.
Like the OP, I’m somewhat annoyed about outstanding loans to siblings that will clearly never be repaid, while my father requires extremely expensive care in a residential facility.
So, if I can offer any advice of value to the OP, I’ll do it.
If the poster is a sock or some other fictional creature, what have I lost? Five minutes?
So it seems worthwhile to me to take the post, and the poster, at face value.
Plenty of people live in section 8 housing and are thankful for it. If you mom qualifies and needs it, she should take advantage of it. You can still help your mom by visiting her and helping her with extra expenses. You should not feel obligated to financially cover your mom’s bad decisions and the thefts committed by your siblings.
She wasn’t an investor; she was a lender. Even if no contract was written, there is a verbal contract, and if nothing else, basic decency would dictate that a borrower make a good faith attempt to repay what he/she is able to, particularly given the current hardship of the lender.