Things that will work
Since this lair action creates "magical darkness", that means that it is suppressed when it overlaps with the effects from antimagic field.
Spells and other magical effects...are suppressed in the sphere and can't protrude into it...While an effect is suppressed, it doesn't function, but the time it spends suppressed counts against its duration.
However, since antimagic field is only a 10 ft sphere (as opposed to the 15 ft sphere area of darkness) there is no way for it to suppress all the magical darkness at once.
Daylight or any spell producing magical light (cast with a 3rd level slot or higher)
Daylight being a source of magical light, will indeed illuminate any area of the darkness that intersects its radius. Given that daylight's radius is 60 ft, it could easily cover up all of the darkness if the positioning is correct.
Jeremy Crawford clarifies this by saying:
If a source of magical light is not a spell of 2nd level or lower, darkness can be illuminated by that light.
However, it will not dispel the darkness since this magical darkness is not created by a spell.
If any of this spell’s area overlaps with an area of darkness created by a spell of 3rd level or lower, the spell that created the darkness is dispelled.
Following the same logic as daylight, faerie fire and other light spells can also illuminate the darkness. However, they must be cast using a spell slot higher than 2nd level or it will be dispelled by the dragon's darkness.
Magic items or any source of magical light not coming from a spell
If any of the effect's area overlaps with an area of light created by
a spell of 2nd level or lower, the spell that created the light is
Any kind of magical light that is not a spell cannot be dispelled by the effect. Thus, it illuminated the magical darkness at whatever radius and intensity is specified by the relevant effect.
Jeremy Crawford has confirmed this logic specifically in reference to the magical item, Sun Blade1:
The sun blade is magic item. It produces light. It is not a spell of 2nd level or lower. It can, therefore, illuminate the area of a darkness spell.
While perhaps not exactly what you want, the warlock's devil's sight ability will give the warlock (but nobody else) the ability to see through the magical darkness created by the dragon.
You can see in darkness, both magical and nonmagical, to a distance of 120 feet.
Things that won't work
Dispel magic only works on spells and the dragon is not casting a spell by using the lair action. See What happens when you target a "magical effect" with Dispel Magic? for more discussion.
1 - thanks @Gandalfmeansme