The most important selection criterion is that the star should be available for observation for a prolonged time.
Because James Webb observes in the infrared, it must hide the Sun (and Earth and the Moon) behind its sunshield, and can hence only observe some 39% of the sky at any given time (source: NASA). Webb needs around three months for its optical alignment, so we need a star that has just entered its field of view$^\dagger$.
In addition to this, we also don't want a star in a field that is too crowded, and HD 84406 is located in a rather isolated region in the NW part of Ursa Major:
Credit: IAU/S&T/Roger Sinnott/Rick Fienberg with my own annotations.
The star should be bright, but probably not too bright (like Betelgeuse), since we don't want to burn MIRI off from the beginning. EDIT: After discussing with several colleagues, damaging MIRI permanently is probably not going to be an issue. Bright sources can however damage the detector temporarily, showing a "ghost image" of the star in subsequent exposures. I've done that myself on a 1.5 m Earth-based telescope, and JWST has a ~20 times larger area, and ~10 times higher resolution, meaning several 1000 times more photons per pixel.
Anyway, as you see there are some important constraints, but many other stars would fit as well; there are quite a lot to choose from.
And the best part: HD 84406 is a magnitude 6.9 star, so while you can't see it with your naked eye, it should be visible in a pair of regular binoculars.
$\dagger$Stars too close to the ecliptic are only available for a more limited time, so a star that is close to the poles is preferred.