What do you mean by "design," BW? When you arrange lines or letters on a page (or bits on a screen) you are, in a sense designing something.
Because an invoice is a component of a contract, it should include certain information: the name of the recipient, the name of the issuer, a date, a reference to the purpose of the invoice / contract under which you're claiming the money, the amount. In NYS you do also need to identify the state sales tax you're collecting, so it's helpful to provide a grand total.
And because it's about money, it's a good idea to reiterate or state terms: payment is due on demand, you'll discount 4% if paid in 8 business days, you'll add a 1.5% service charge cumulative per month after 30 days, and so forth. (If they're in effect of course) It may be a good idea to include your TIN or attach a W9. You may also want to include account information for direct deposit.
It's also important to make sure that each element is obvious to the client or their accounts payable person. You don't want them paying you $7.18 if that was actually the date of the invoice.
And, if you're concerned about branding your business--if you have a logo, you use a specific background or watermark, typefont and size, and so forth in your communications, then you should use that on your invoice, too.
All these are components of a design. Does it need to be fancier? not if you don't want it to be.