This isn't the "I'm looking for a technical cofounder" type of paying with equity.
This is the entrepreneur looking to get a good deal on the project. In their mind, you can cut your rate in half and take part of the company as equity. The numbers change, but the idea stays the same.
It's usually their money, and they want to get the most bang for their buck. It keeps your interest in the success of the project, and everyone on the same goal. And it's a terrible idea.
Let's imagine a $20,000 project, and you accept 50% of the payment in equity instead of cash. You're out, $10,000, for some amount of equity. Is it $10,000 of the future value of the company, after it's made it? Or is it $10,000 of the current value, which is completely unknown?
Maybe you say no to equity (smart), but say yes to 33% of all sales, much like a sales commission. So you get $3 of every $9 per month subscription.
Either way, you have absolutely no control in the decision and direction of the company. You're still a contractor, and the client can decide to not launch the product. Which has happened to me. Twice.
The client gets to the end, and discovers the project will take more of their time than their originally thought. A project doesn't run themselves, which you know, but the client may not have fully realized.
In this project, you worked for $10,000 and left $10,000 on the table. That money is gone and you'll never see it. 33% of 0 is still 0.
Take the cash, leave the equity.
How to say no, without saying no
In the same $20,000 project, with a $10,000 reduction, you can say no without saying no by asking the client for an audit to come up with the equity your $10,000 will take up.
"I'd like to be on the same page, with the same goals as you are. Unfortunately, we're in a bit of a pickle with this. I'm uncertain what the $10,000 of equity will be worth of the company; we can both agree it's likely worth way more today than $10,000 later.
"If you can produce a financial audit that values your company at a certain number, we'll be able to determine the amount of equity the $10,000 reduction provides me. Unfortunately, this transaction will likely take just about that much in legal fees to close. Is that something you'd like to pursue?