If your name field is the 4th column (as expected for BED format), you would do the following. This assumes your BED file is sorted by chromosome and then by start position.

bedtools merge -i file.bed -c 4 -o collapse

If the names are in the fourth column, here's one way to get a unique list of names that map across merged elements, where the input is sorted:

$ sort-bed unsorted.file.bed > file.bed
$ bedops --merge file.bed \
    | bedmap --echo --echo-map-id-uniq --delim '\t' - file.bed \
    > answer.bed

Mapped IDs will be in the fourth column of answer.bed.