If your institution has a license, I think IPA is the best option (but it isn't free).
IPA has a mix of known and predicted targets. You can download the predicted targets for free (http://www.targetscan.org/). Many of the known interaction may come from IPA's proprietary database, but I think you can also find known interactions from tarbase (http://diana.cslab.ece.ntua.gr/tarbase/).
Once you have text files with the miRNA-mRNA interactions, you can search for which interactions fit your data (based upon genes with with opposite fold-change / log2ratios and/or a negative correlation between miRNA and mRNA expression levels).
You also can use starBase ( http://starbase.sysu.edu.cn ).
starBase Platform is designed for deciphering Pan-Cancer Networks of miRNA-mRNA, miRNA-lncRNA, miRNA-ncRNA and protein-RNA by mining clinical and expression profiles of 14 cancer types (>6000 samples) from The Cancer Genome Atlas (TCGA) Data Portal. It explore the negative correlation between miRNA and mRNA expression levels using expression profiles ( http://starbase.sysu.edu.cn/targetSite.php ).
If you're still interesting in such an analysis, there is a new version of MMIA, which offers additionally analysis for NGS data.
http://omictools.com/biovlab-mmia-ngs-s6768.html