An HDMI to DisplayPort adapter tends to run around 80 USD. It basically presents itself as a monitor to the graphics card, then copies the HDMI frames to DisplayPort frames for the monitor. Usually they only go up to 1920x1200, so you might have to look hard to find one that is full resolution if that's something you need. There should be no theoretical issues with it, but make sure that HDMI is the computer side of the adapter, and that display port is the display side. They are sometimes labeled misleadingly.
If you are building a PC, then it's probably new enough to have USB 3.0. After someone else was talking about not having a second display port, I eventually got to thinking that a USB Graphics card would be an interesting idea. You can find a lot of different models of small USB graphics cards online, and many of them have a display port output. They are nothing for gaming, but they work fine for destop use. They are usually a little under 100 USD.
Although for those prices, you could get something like the GTX 750 which uses a much faster link than USB 3.0.
