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There are many many studies that prove it works, however, the remedies cost virtually nothing compared to the big multi-billion dollar pharma industry. The issue is that we don't understand the mechanism for how it works - we still haven't figured out gravity, but no one calls it "witchcraft".
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Albertini H, Goldberg W, Sanguy B, Toulza CL. Homeopathic treatment of dental neuralgia by Arnica and Hypericum. Journal of the American Institute of Homeopathy, 1985, 3, 126-129. Carried out at the Faculty of Medicine of Marseilles, this placebo controlled study was designed to determine the effectiveness of Arnica 7C and Hypericum 15C for people suffering from dental neuralgia. 60 people received either 4 pilules of Arnica alternated with 4 pilules of Hypericum every 4 hours or placebo administered in the same way. Pain levels were assessed over 3 days from the beginning of the trial. From this assessment it was found that 12 of the 30 people who received the placebo had a positive response to this intervention, and 23 of the 30 people given the homeopathic medicines responded positively to these.

2. Alibeu JP, Jobert J. Aconite in Post-Operative Pain and Agitation in Children, Pediatrie, 1990, 45, 7-8, 465-6. In this double blind placebo controlled trial, 50 children suffering from post-operative pain and agitation were given either Aconite or placebo. The effects of Aconite were significantly better than placebo

3. Ammerschlager H., Klein P., Weiser M., Oberbaum M. Treatment of inflammatory diseases of the upper respiratory tract – comparison of a homeopathic complex remedy with xylometazoline. Forsch Komplementarmed Klass Naturheilkd. 2005, Feb, 12, 1,:24-31. In a multi-centre, controlled, cohort study, 739 people suffering from rhinitis or sinusitis were given either xylometazoline (a drug commonly used for nasal congestion, sinusitis and hay fever) or a homeopathic combination. Both forms of therapy provided similarly effective results