Bootstrapping as a way of learning about the population I have a small sample from an unknown, possibly non-parametric population. I would like to create a new, different sample from the population based only on what I can extract from the small sample I have.
Is there a way to do it, given that my sample is small? Specifically, can I resample from that sample using bootstrapping and hope I can get closer to the "true" mean and variance of the population? (Let's assume for a moment these two parameters are enough to characterize the population.)  
 A: No, you cannot. Bootstrap isn't magic: it cannot create new information.  If you want/need a new, different sample from the population the only way to get that is to sample from the population!
Bootstrapping and resampling is a way to analyze the information in your sample.  Its grounding, apart from its intuitive appeal, is in large sample theory, that is, approximations based on a large sample size.  So if your sample is very small, it might be that bootstrapping is not a good way to analyze it. You haven't given us enough details and context to say much more. 
A: What bootstrapping CAN give you an estimate of is, for example, how much your estimated mean would vary across samples of the same size as you have. This is however not the same thing as the variance in your data (variance of the mean vs variance within between samples). Especially in the case of the mean, bootstrapping won't generally do anything at all, you can take as many bootstrap samples as you want, the mean won't change in the expectation.
