Saying "There's a lot of options" is incorrect because of the misunderstanding of the noun in the sentence.
"There is" (There's) is for singular nouns, and "There are" is for plural nouns.
"A lot of..." is a phrase that either describes a singular noun or plural noun and is usually mistaken for the noun of the sentence.
What would the sentence sound like if "...a lot of..." was taken out? ("There's options" or "There is options."
Immediately, you would recognize the wrong verb usage; it should be "There are options."