If you need the article strictly for the noun phrase as written (without adding "for"), it would still be because "home remedy" begins with a consonant sound: "a home remedy blocked tear duct" But that construction is grammatically awkward. The natural and correct way is: "a home remedy for a blocked tear duct"
For the phrase , the most appropriate article is "a" : "A home remedy for a blocked tear duct" (You also need the preposition "for" and an article before "blocked tear duct" — typically "a" as well, unless referring to a specific case.) home remedy blocked tear duct