What about something on these lines using python and the Levenshtein module: (not tested)

import Levenshtein as ls
import itertools

mirna= ['ACTG', 'ACG', 'TTTT']

# Get edit distance and ratio between all the pairs of miRNA:
for x, y in itertools.combinations(mirna, 2):
    print(x, y, ls.distance(x, y), ls.ratio(x, y))

Output:

('ACTG', 'ACG', 1, 0.8571428571428571)
('ACTG', 'TTTT', 3, 0.25)
('ACG', 'TTTT', 4, 0.0)

EDIT: Use itertools to generate pairs of miRNA.

You can generate a blast database and align align all mirna sequences against it in a second pass. The choose some criteria for cutoff (e.g. top X hits with E-value > Y) and build lists of homologs for each mirna sequence. Then I would recommend building a cytoscape graph. It should be very easy, just input graph as "mir1\tmir2" and edge characteristics as "mir1 (pp) mir2\t2\t1" with header "interaction\tmismatches\tindels", push it to cytoscape and visualize using edge weights. The resulting picture should be awesome :)

If I understood the question properly, you could probably use closestBed while ignoring the same overlapping ones

closestBed -a mirbase.bed -b mirbase.bed -io

-io option removes overlapping ones, and there are other options available that u can play with.

BTW, what did you mean by edit distance (I mean in biological term)?