The like operator applies to a text, not to an array, so you need to find a way to convert your array into text before applying the like operator. To do so in sql you can use array_to_string() function, see the manual.
For example :
select array_to_string('{"name_1", "name_2", "name_3"}' :: text[], ' ') LIKE '%name_2%' returns true.
The first character % is mandatory because the searched word may be at any place in the string which results from the array.
Then you can add the separator specified in the array_to_string() function before and after the searched word (LIKE '% name2 %') in order to exclude the results like somename_2.
For example :
select array_to_string('{"name_1", "name_2", "name_3"}' :: text[], ' ') LIKE '% name_2 %' returns true
but
select array_to_string('{"name_1", "somename_2", "name_3"}' :: text[], ' ') LIKE '% name_2 %' returns false.
Last but not least, for the seprator, you can specifiy a non printable character like chr(12) or a set of characters instead of the space character in order to secure the results of the query.