From my experience, the greatest objection potential converts have to libertarianism is its non-negotiable advocacy of abolishing government "safety nets." People just seem to think it's downright callous to oppose public health care for children, food stamps for out of luck families, and the whole welfare apparatus.
In seeking to persuade such individuals, I offer you this succinct argument by Mary Ruwart:
...although refusing to help others might not be very loving, pointing guns at our neighbors to force them to help those in need was even less so. Honoring our neighbor’s choice was more loving than the forcible alternative. If people needed helping, I should expend my energy to offer that help, rather than forcing others to provide it.