Use the Variant Effect Predictor and select the LoFtool, available both in the online interface and Perl script as a VEP plugin. LoFtool gives you 'a rank of genic intolerance and consequent susceptibility to disease based on the ratio of Loss-of-function (LoF) to synonymous mutations for each gene in 60,706 individuals from ExAC).

Synonymous variants are not significant at the protein level - you should look at DNA/RNA level effects. Look for motifs affected, binding sites disrupted, cryptic splice sites created, etc.

Tools should not the first thing on your list - the concept of consequence/effect should. Your biological question will drive the kind of effects relevant to your analysis, and the kind of effect you're looking for will determine what tool would be appropriate.

You could also try something like SilVA

http://compbio.cs.toronto.edu/silva/

Identification of deleterious synonymous variants in human genomes.