Any airport does not qualify. Most of those airports have "A-NA" on
the
chart, which stands for "Alternate - Not Authorized".

Actually, that's not quite right. The A-NA refers to the approach, not
the airport. It is quite common to have some approaches to a given
airport marked A-NA, and others not so marked. In that case, the
'best' approach among those not marked A-NA is used to determine
alternate minima for that airport. If all approaches are marked A-NA,
then the only remaining approach is a visual. The alternate minima for
a visual are weather sufficient for descent from MEA to a landing under
basic VFR. Those minima are always authorized, so any airport is an
authorized alternate - the only question is what the alternate minima
will be. There are airports where the alternate minima require 10,000'
ceilings (mountain MEA's and valley airports will do that) but they are
still authorized.

I agree with you that there are not any airports within a short
distance of DAY that would be useful alternates in conditions that
would actually require you to file an alternate for DAY.

Michael