When we say "I will show you", we can't predict if this "you" is singular or plural, therefore both "eu lhe mostrarei" and "eu lhes mostrarei" are correct. But remeber it is correct only if you choose to use pronoun "você". If you choose pronoun "tu (pl. vós)" answers will be "eu te mostrarei" (sing.) and "eu vos mostrarei" (pl.).
But let's go further and do the opposite. How would you say "eu lhe mostrarei (isso)" in english? The "lhe" is indirect object of pronouns "você", "ela" and "ele". So without any context to background it, it could be "I will show you/her/him (this)".
As you can see, "lhe" is an ambiguous pronoun. That's why when we need to be clear and specific we use preposition plus pronoun. Instead of "eu lhe mostrarei" we say "eu mostrarei a você/ a ela/ a ele". Instead of "eu lhes mostrarei" we say "eu mostrarei a vocês/ a elas/ a eles".
Last tips. Pronoun "eu" can be omitted here: "mostrarei a você" is enough for any portugues speaker. Choose the preposition you like: "mostrarei a você" or "mostrarei para você" are the same thing. In PT-BR we're not big fans of using verbs in the future tense: "vou mostrar para você" is what you hear is the streets and it means the same. And "para" is often contracted to "pra", so: "vou mostrar pra você". There you go.