I put egg production first, but as mine are truly backyard chickens, other factors came into play. I have children so their preferences come into play as well.
The first chickens were two EE's, a RIR and a RSL. The RSL was an egg laying machine, and very friendly; but ended up becoming an internal layer. These chickens were picked for egg production, egg color, and when we chose the EEs we picked the most colorful ones we could find.
The next ones were silkies, RIR and BO hatching eggs. The silkies were chosen for looks, the BO for their winter laying skills, and the RIR for general egg production. A SLW chick was added to the mix entirely for looks. The RIR cockeral we hatched was almost immediately rehomed due to a general bad attitude. We kept the silkies for about a year, but the rooster's constant crowing, and nasty habit of attacking my younger child made him an undesirable. He and his girls were rehomed.
We tried some Japanese bantams, but they were too flighty and wouldn't return to the coop. I won Marans hatching eggs so they were added to the flock. We also have one BO and a BO/RIR from our own crosses. They went into the incubator as test eggs.
Now we have hens: 4 EEs, 3 BO, 2 Marans, 1 SLW, 2 RIR, 1 RIR/BO cross
Roos: BO and Marans...they stay because they are beautiful. I don't need them to breed in my backyard flock, and the BO's idea of flock protection is to get to the highest place he can and alert all the hens to danger.
So in order:
Egg production but with the notion that the egg laying machine was not as healthy as birds with lower production
Personality/attitude: I don't have the time, energy or patience for nasty or flighty birds.
Looks, because eye candy is important
Egg color because that is just fun
If I was farming for myself I would also be interested in dual purpose birds that breed themselves. In my flock that would mean keeping the buff orpingtons and Marans. I would also have OEGB simply because I love the way the look, and the variety of colors these birds come in.