The Dragonmark Feat from Unearthed Arcana: Eberron says:
You gain the ability to innately cast spells and cantrips, as
summarized in the Dragonmark Benefits table, using the spellcasting
ability listed under the Ability column. You cast each spell at its
lowest level. Once you cast a given spell this way, you must finish a
long rest before you can cast it innately again. You must still expend
any material components.
Every dragonmark gives the character access to a first, second and third level spell, as well as a cantrip. The second and third level spells can only be cast if the character is of high enough level (, 5th and 9th, respectively).
Usually you can cast the cantrips you know infinitely often. But the way this feat is worded seems to imply, that the cantrips gained from dragonmarks are treated exactly like the other spells, which means their use is limited to once per day. I’m am not sure if I interpret this correctly, if this is intended this way, or just worded weirdly.
Best Answer
You can only cast them once per long rest
As you quoted above, you can only cast each spell once per long rest. A cantrip is still a spell (just at 0 level, which would be the lowest level you cast it as) and therefore falls under the same requirement.
Counter-examples
Dragonmark differs in its language and requirements from things like the Magic Initiate Feat or The cantrips gained from a certain race (like High-elf.) Those simply add that cantrip to your current list and do not have a limitation on uses/day.