As one might suppose, the quality of emergency queens is not simply a
matter of the bees raising queens from a bunch of different eggs and
letting the first (and worst) come out and rule. While they do use eggs
and larvae of different ages, they then proceed to destroy a lot of them,
suggesting that they are "evaluating' them in some fashion. According to
work by Dave Tarpy, et al., bees seem to prefer queens raised from older
eggs versus either those from younger eggs or older larvae.

(quoted material):

Selection of high-quality queens by the workers during
queen development has been demonstrated by Hatch et al.
(1999), who found that during emergency queen rearing
workers preferentially destroyed queen cells built from
older worker larvae. Workers destroyed 53% of the queen cells
that they initiated.

Despite selective behavior by the
workers during queen rearing, considerable variation in
quality exists among newly emerged adult queens. This
variation in quality among queens gives workers the
opportunity to benefit by selecting high quality queens
that are fully developed, when the decision will be most
accurate.

Honeybee workers interact with queens extensively
following queen emergence and may affect
the outcomes of queen duels in favor of high-quality
queens.

SEE:
The influence of queen age and quality during queen replacement in
honeybee colonies DAVID R. TARPY, SHANTI HATCH & DAVID J. C. FLETCHER

Worker regulation of emergency queen rearing in honey bee colonies
S. Hatch, D.R.Tarpy, and D.J.C. Fletcher

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