My sister and I are closest to one cousin among all of our relatives. Is there anything we can do to help him immigrate?
You cannot directly petition for a cousin. So, you have to find another way to assist your cousin.
As my colleagues pointed out, you cannot petition for a cousin to come to the US on a family based immigrant visa. Your cousin would need a US Citizen spouse, parent, child, or sibling or a permanent resident spouse or parent to petition for them. The only way you could help out is if the petitioning relative did not meet the income requirements to support your cousin, then you could offer to "sponsor" him.
If he doesn't have one of these relatives, he may want to look into an employment based visa and his eligibility for one of these visas is going to depend on his profession, education, skills, etc.
If he doesn't have the necessary skills or degree necessary to pursue an employment based visa and he is thinking about furthering his education, he can always apply for a student visa. If he needs help with the financial requirements, you could help him with that as well.
If you have more detailed questions, talk to an immigration attorney.