Before I get to your question, let me give you a tip: always aim settle questions of payment before the work happens. It is ten times easier to agree on a price beforehand, and having done that doesn't stop you from changing it by mutual agreement later.

The problem with paying cash is pretty obvious: you don't have a lot of it. The problems with paying equity are subtler. The first one is that early-stage equity is extremely hard to value. A second is that equity transactions require a lot of paperwork. Third is that entrepreneurs tend to value their equity much higher than other people would; if not, they wouldn't be starting the company. And fourth, people like designers are rarely expert in valuing businesses or the customs of of startup equity valuation. In the past, I've both given and received equity compensation, and it's a lot more of a pain than I expected.

In the future, what I think I'd try is convertible debt. That is, I'd talk with the designer and agree on a fair-market wage. E.g. 100 hours x $100/hr = $10k. The next time we take investment, the $10k turns into stock at whatever price we agree with our investors, plus a discount because he was in before the investors.

Note, though, that this will increase your legal costs and your deal complexity, so I'd personally only do this for a pretty significant amount of work. And I'd only do it for somebody I trusted and respected enough to have them around for the life of my business.