There is evidence of analgesic effects of menthol in scientific literature. It has been studied in humans and has shown to be superior to ice in delayed onset muscle soreness; in a placebo-controlled, triple-blind, cross-over clinical study menthol-based gel acutely reduced pain in subjects with carpal tunnel syndrome symptoms. In mice blockage of voltage gated Na-channels of dorsal root ganglion neurons has shed light on potential mechanisms of analgesic action.
Here are the key parts of these studies:
A comparison of topical menthol to ice on pain, evoked tetanic and voluntary force during delayed onset muscle soreness has shown that:
Compared to ice, the topical menthol-based analgesic decreased perceived discomfort to a greater extent and permitted greater tetanic forces to be produced. [...] Tetanic force changes illustrated a significant main effect for the treatments (p<0.05; ES=1.1) with the menthol based topical analgesic allowing 116.9% greater tetanic force (89.4 N ± 60.7) output than the ice treatment (41.2 ± 43.6). [...] There was a significant (p=0.025; ES=1.2) difference in soreness perception with the VAS scale between the application of ice and the menthol based topical analgesic. Soreness perception was 63.1% less with application of the topical analgesic (1.1 ± 0.4) compared to the ice (3.1 ± 1.7).
In this study menthol was applied as 3.5% gel (Biofreeze®) without substantial force or rubbing during application.
Aside from cooling sensation attributed to activation of TRPM8 channel in this and other studies, another study (Menthol pain relief through cumulative inactivation of voltage-gated sodium channels) has tested the hypothesis that menthol could block voltage gated Na-channels:
The results indicate that menthol inhibits Na+ channels in a concentration-, voltage-, and frequency-dependent manner. Menthol promoted fast and slow inactivation states, causing use-dependent depression of Na+ channel activity. In current clamp recordings, menthol inhibited firing at high-frequency stimulation with minimal effects on normal neuronal activity. We found that low concentrations of menthol cause analgesia in mice, relieving pain produced by a Na+ channel-targeting toxin. We conclude that menthol is a state-selective blocker of Nav1.8, Nav1.9, and TTX-sensitive Na+ channels, indicating a role for Na+ channel blockade in the efficacy of menthol as topical analgesic compound.
Acute Effect of Topical Menthol on Chronic Pain in Slaughterhouse Workers with Carpal Tunnel Syndrome: Triple-Blind, Randomized Placebo-Controlled Trial:
Topical gel containing menthol led to a 31% (1.3 point on 0–10 VAS) acute reduction in chronic pain associated with carpal tunnel syndrome, and the absolute change in pain symptoms between topical menthol and placebo was 1.2 corresponding to a moderate effect size
In official monographs one can find predominantly Peppermint oil (Menthae piperitae aetheroleum), but since literature states that menthol is it's main ingredient (30 - 55% [WHO]) we could assume that it plays a role in the effects of the oil.
Traditional use, indication 2:
For the symptomatic relief of localised muscle pain
Uses supported by clinical data
Internally for symptomatic treatment of irritable bowel syndrome (15-20), and digestive disorders such as flatulence and gastritis (21-23). Externally for treatment of myalgia and headache (21, 24-27)
The Commission E approved the internal use of peppermint oil for spastic discomfort of the upper gastrointestinal tract and bile ducts, irritable colon (in enteric-coated capsules), catarrhs of the respiratory tract, and inflammation of the oral mucosa; and external use for myalgia and neuralgia.
ESCOP [...] Its external use is indicated for coughs and colds, rheumatic complaints, pruritus, urticaria, and pain in irritable skin conditions (ESCOP, 1997).
*To show the lack of bias: not an official monograph, but a respected resource, PDR for Herbal Medicines lists the cutaneous use of peppermint oil as an analgesic in myalgia and neuralgia as "unproven uses".