Bookkeeping and double-entry accounting is really designed for tracking the finances of a single entity. It sounds like you're trying to use it to keep multiple entities' information, which may somewhat work but isn't really going to be the easiest to understand. Here's a few approaches:
Approach #1: The books are for "your" money, not the kids'
In this approach, the books are entirely from your perspective. So, if you're holding onto money that "really" belongs to your kids, then what you've done is you're taking a loan from them. This means that you should record it as a liability on your books. If you received $300, of which $100 was actually yours, $100 belongs to Kid #1 (and thus is a loan from him), and $100 belongs to Kid #2 (and thus is a loan from her), you'd record it just that way.
Account Debit Credit
----------------------- -------- --------
Income:Garage Sale $100.00
Liability:Kid #1 $100.00
Liability:Kid #2 $100.00
Assets:Petty Cash $300.00
Note that you only received $100 of income, since that's the only money that's "yours", and the other $200 you're only holding on behalf of your kids. When you give the money to your kids or spend it on their behalf, then you debit the liability accordingly and credit the Petty Cash or other account you spent it from.
If you wanted to do this in excruciating detail, then your kids could each have their own set of books, in which they would see a transfer from their own Income:Garage Sale account into their Assets:Held by Parents account.
Approach #2: All assets have subaccounts
For this, you just apportion each of your asset accounts into subaccounts tracking how much money each of you has in it.
Account Debit Credit
----------------------- -------- --------
Income:Garage Sale $300.00
Assets:Petty Cash:Mine $100.00
Assets:Petty Cash:Kid #1 $100.00
Assets:Petty Cash:Kid #2 $100.00
This lets you treat the whole family as one single entity, sharing in the income, expenses, etc. It lets you see the whole pool of money as being the family's, but also lets you track internally some value of assets for each person. Whenever you spend money you need to record which subaccount it came from, and it could be more challenging if you actually need to record income or expenses separately per person (for some sort of tax reasons, say) unless you also break up each Income and Expense account per person as well. (In which case, it may be easier just to have each person keep their entirely separate set of books.)
Approach #3: Equity accounts, but the books from "your" perspective
I don't see a whole lot of advantages, but I'll mention it because you suggested using equity accounts. Equity is designed for tracking how much "capital" each "investor" contributes to the entity, and for tracking a household it can be hard for that to make a lot of sense, though I suppose it can be done. From a math perspective, Equity is treated exactly like Liabilities in the accounting equation, so you could end up using it a lot like in my Approach #1, where Equity represents how much you owe each of the kids. But in that case, I'd find it simpler to just go ahead and treat them as Liabilities. But if it makes you feel better to just use the word Equity rather than Liability, to represent that the kids are "investing" in the household or the like, go right ahead.
If you're going to look at the books from your perspective and the kids as investing in it, the transaction would look like this:
Account Debit Credit
----------------------- -------- --------
Income:Garage Sale $100.00
Equity:Kid #1 $100.00
Equity:Kid #2 $100.00
Assets:Petty Cash $300.00
And it's really all handled in the same way an Approach #1.
Approach #4: The family as its own separate entity
If on the other hand, you really want the books to represent "the family", then you'd need to have the family's books really look more like a partnership. This is getting a bit out of my league, but I'd imagine it'd be something like this:
Account Debit Credit
----------------------- -------- --------
Income:Garage Sale $300.00
Assets:Petty Cash $300.00
That is to say, the family make the sale, and has the money, and the "shareholders" could see it as such, but don't have any obvious direct claim to the money since there hasn't been a distribution to them yet. Any assets would just be assumed to be split three ways, if it's an equal partnership. Then, when being spent, the entity would have an Expense transaction of "Dividend" or the like, where it distributes the money to the shareholders so that they could do something with it.
Alternatively, you'd just have the capital be contributed,
Account Debit Credit
----------------------- -------- --------
Equity:Parent $100.00
Equity:Kid #1 $100.00
Equity:Kid #2 $100.00
Assets:Petty Cash $300.00
And then any "income" would have to be handled on the individual books of the "investors" involved, as it would represent that they make the money, and then contributed it to the "family books".
This approach seems much more complicated than I'd want to do myself, though.
