In common with nearly all Tolkien's dwarf-names, Gimli comes from a Mannish tongue, based on Old Norse. Unusually, Tolkien himself seems uncertain on the original meaning of Gimli. In his Letters, he writes 'its meaning seems to have been "fire"' (The Letters of J.R.R. Tolkien, No 297, dated 1967).
In Norse mythology, Gimli was not the name of a person, but of a place: a great golden-roofed hall in Asgard where the righteous were taken after death. A golden-roofed hall appears in Tolkien's work, too, of course - the Golden Hall of Meduseld at Edoras - though it is unclear whether there is a connection.
In Adûnaic, gimli means 'stars' (gimil, 'star', appears in names such as Gimilkhâd) but this is probably no more than a coincidence.