How do honeybees pick the queen?

They don't.

There can only be one queen in a hive. If a hive becomes overcrowded the queen will lay eggs in special queen cells and the nurse bees will look after these and feed them exclusively with royal jelly. That is what determines the larvae will develop into queens. Before these larvae pupate and mature the queen will leave the hive with about half of the workers. That is a swarm.

When the first adult queen emerges from the brood cell she will search round for the other queen cells, and each one she finds she will sting through the cell wall to kill the developing queen inside. If two queens emerge at the same time, they will fight until one manages to sting the other to death.