So long as you don't hate what you are doing, I'd say the price is somewhere in the neighborhood of $100-200 year of income to be worth the bookkeeping. I'd only say more than that if you have a ridiculously complex tax situation, you have an irrational hatred of filling out a few forms once a year, or if you just have such a stupidly large amount of money that even having a few hundred dollars a year to donate to people in desperate need just doesn't mean anything to you. Or if you are under special income limits and just a few dollars of income would put you in a bad situation (like a loss of medical benefits, etc).
The reason is actually quite simple: the taxes aren't that really that hard or time consuming. I've handled three self-employment businesses in my life, and unless you are trying to itemize every last dollar of business deductions and expenses, or you really want to scrape out every last cent from minor deductions that require considerable extra paperwork, it's a few extra forms on your taxes. Most of the extra taxes are as a percentage, so it reduces the benefits, but really not by much.
You don't have to make it extra complicated if the extra complexity doesn't give you a big payoff in benefit.
I would suggest you pick the simplest imaginable possible system for accounting for this, so that you might only spend an extra few hours per year on the books and taxes. Don't keep $10 sheet music receipts if you feel it's a burden to try to itemize expenses, etc. Instead, the decision should be if you (or in this case your wife) would enjoy doing it, and bringing in money can just be nice in it's own way. I'd suggest she keep some out for little extra niceties, earmark some for feel-good charitable giving, and then of course sock away the rest.
Don't let extra income be an unnecessary burden that prevents you from getting it in the first place.