Have a look at the msigdbr R package. It easily converts mouse gene names to human homolog equivalents.

Jackson Lab has the Vertebrate Homology resource for converting between homologous human and mouse genes (most of the time, it's just adjusting the case).

Update: you can also try msigdbr which already solved the multi-species conversion issue (there are a number of caveats to the process that are not very obvious): MSigDB for Multiple Organisms in a Tidy Data Format

Generally I use Ensembl Biomart (but you can substitute your favorite gene ID conversion tool here) to convert all of the gene names in each gene signature in the *.gmx or *.gmt, from human to mouse. I then use the mouse IDs to make a custom *.gmx or *.gmt file.

It's convenient to do this in R if you know how but not necessary, it works perfectly well do to copy-paste into the web interface.

Word of warning: look out for genes that don't convert using automated tools. Often you will come across genes that actually do have mouse orthologs but are not annotated in Biomart. You can manually add these genes to the signatures by a quick Google search.

Here is a solution using biomaRt (in R) from Ensembl which creates a table that lists all mouse genes and the respective human orthologs. Credits mainly to Dave Tang's blog where the idea is from.

library(biomaRt)

Mouse2Human <- function(MouseGenes){

  human = useMart("ensembl", dataset = "hsapiens_gene_ensembl")
  mouse = useMart("ensembl", dataset = "mmusculus_gene_ensembl")

  genesMousetoHuman = getLDS(attributes = c("ensembl_gene_id","mgi_symbol"), 
                             filters = "mgi_symbol", 
                             values = MouseGenes , 
                             mart = mouse, 
                             attributesL = c("ensembl_gene_id", "hgnc_symbol"), 
                             martL = human, 
                             uniqueRows = TRUE)

  colnames(genesMousetoHuman) <- c("Mouse.Gene_ID", "MGI", "Human.Gene_ID", "HGNC")

  return(genesMousetoHuman) 

}

## Get mouse genes
mmusculus_genes <- getBM(attributes = c("ensembl_gene_id", "mgi_symbol"),  
                         mart = useMart("ensembl", dataset = "mmusculus_gene_ensembl"),
                         useCache = FALSE)

## create the conversion table
Mouse2HumanTable <- Mouse2Human(MouseGenes = mmusculus_genes$mgi_symbol)

Output:

> Mouse2HumanTable
         Mouse.Gene_ID           MGI   Human.Gene_ID           HGNC
1   ENSMUSG00000070979        Actl7a ENSG00000187003         ACTL7A
2   ENSMUSG00000069911       Insyn2b ENSG00000204767        INSYN2B
3   ENSMUSG00000028173           Wls ENSG00000116729            WLS
4   ENSMUSG00000094356      Igkv8-28 ENSG00000211598        IGKV4-1
5   ENSMUSG00000104769      Igkv8-34 ENSG00000211598        IGKV4-1
6   ENSMUSG00000076581      Igkv8-26 ENSG00000211598        IGKV4-1
7   ENSMUSG00000095794      Igkv6-17 ENSG00000211598        IGKV4-1
8   ENSMUSG00000064370       mt-Cytb ENSG00000198727         MT-CYB
9   ENSMUSG00000022099          Dmtn ENSG00000158856           DMTN
10  ENSMUSG00000033717        Adra2a ENSG00000150594         ADRA2A
11  ENSMUSG00000022016        Akap11 ENSG00000023516         AKAP11
(...)

I tried GSEA with mouse data both by mapping the homologs using a proper method (ensembl/biomaRt as suggested here), and also by simply converting sentence case gene names (i.e. Pdcd1) to upper case (i.e. PDCD1). In my experience, these approaches work similarly.

Using uppercase letters, for the most part, works because human gene names are simply a capitalized version of mouse genes. Although this "conversion" may result in losing a few genes, since you are looking at a constellation of genes in GSEA, I don't think it is a big problem.