Yes. For the
fifth generation version NUC you've linked to, a passive mini-DP to HDMI cable would work.
One suggestion, for fifty CAD more would you consider latest
seventh generation version? The primary difference is slightly faster CPU and upgraded output options. Use the
NUC Wikipedia page to help sort out the NUC model lineup. The seventh generation has an HDMI port on the left and a highly versatile
Thunderbolt 3 port on the right.
So, you'd need to adjust cables. TB3 is nice, but because it's so versatile it can be confusing. It carries all sorts of signals in one, so adapters can get ridiculous.
They do make simple simple
TB3 to HDMI adapters that will work on a seventh gen.
So, good luck! One last thing, does your friend have strong feelings about how much RAM / HDD space they want? Would they care to have less CPU/RAM and something like a giant 1TB HDD? I'm always a bit surprised by this, but some folks want it. In the end, it's their PC, so get a feel for what they think is important and cater to it. Perhaps offer several NUC options, and let them pick? You'll get buy-in and both parties will be happier in the long run.
Let us know what you end up getting.