You can start narrowing down the choice of reasonable likelihoods with any constraints. If you know the data is going to be purely positive, then you can rule out distributions that assign data to the entire real line. For discrete counts you can use the Poisson or negative binomial.
One line of reasoning for likelihood selection is that you should use the maximum entropy distribution that satisfies any constraints you have. For a random variable with finite mean and variance, the max entropy distribution is the normal distribution. Can you supply any constraints on the possible values of the observed data?