There may not be a regulatory element involved. The point mutation may affect how well the transcript is translated due to elimination of a rare codon, elimination of an mRNA structure that interferes with translation, or some other effect of mRNA sequence on translation efficiency.
Shapiro has already given the explanation. I just wanted to add, if the point mutation changes a very crucial and conserved amino acid residue then, it might affect its folding also. Misfolded protein will be degraded as it expresses to avoid formation of toxic aggregates inside the cell. Hence, if you are seeing the end product in whole cell lysate by western blot, then you may see very little or no expression at all. Although your protein will be expressed but will get degraded also, giving the impression that there is a problem with the level of expression. Moreover, as Shapiro rightly said it might alter the transcription and translation efficiency.
1. You can check that by a real time PCR followed by western blot to identify where exactly is the problem.
2. If you rule out transcription (if you get the mRNA in PCR), then you can make whole cell lysates at different time points post transfection to check if you can capture it before degradation. Also, you might use some proteasome inhibitor and prohibit the proteolysis. If you then get a blot for your protein it would mean that translation was fine but it was getting degraded by the machinery as response to misfolding of your protein.
Agreeing in general with Shapiro and Saha, a point mutation in an open reading frame per se does not alter protein expression. Then your effect is (in my mind) indirect.
I think you do not specify whether you are working with the expression in bacteria or in an eukaryotic cell. And what type of protein is?: a receptor, collagen, a metabolic enzyme.... All these details are needed to propose likely alternatives to be easily testable.