Practically speaking, there is no reason to beat your friend to the mineral, because you both end up with each other's minerals at the end of the expedition.
If your goal is to earn crowns, then you should go for the biggest mineral you can find. When choosing between two minerals of the same size, you should favor yellow, green, and red over blue and purple, because the former often sell for 4 crowns per while the latter rarely sell above 3 crowns per. (This is based on multiple months of obsessively observing mineral prices.)
Alternatively, if your goal is to influence stratum themes, then research the theme you want and pick up minerals accordingly. I recommend saving up for a long time, or joining a guild that's into gate construction.
If you don't care about such practical goals...then choose the mineral that goes well with your armor/costume, and always beat your friend to it? :)
Biggest usually, except for the one I have most of (blue).
Though there are many times my I pick up smaller minerals least collected color (Green). It's an odd complex of mine to make the quantities of all match, and failing bad since while 4 are in the upper 2800s/low 2900s, green's the odd one out at 2700s.