There are many synonyms for "give someone a try," depending on the context and tone. Alternatively, one could say "give someone an opportunity," "give someone a chance," "take a chance on someone," "test someone out," "put someone to the test," "give someone a shot," "give someone a break," "offer someone a prospect," "afford someone a possibility," "present someone a challenge," "allow someone an opening," or "grant someone a shot." Each of these phrases captures the sentiment of giving someone an opportunity to prove their worth or demonstrate their abilities. Some of these synonyms may have slightly different connotations or implications, but they all share the underlying idea of offering a chance to someone.