All mulching materials have advantages and disadvantages. Research did not yield any documents that make a watertight case for or against any of them, including crushed rock, lava, etc. Here are some frequently mentioned concerns:
1. Rock mulch such as limestone or marble chips may raise the pH of the underlying soil. This can lead to poor growth and chlorosis in some plants, especially rhododendrons, azaleas, blueberries, and certain evergreens.
2. Rock mulches warm the soil rapidly and hold heat longer than inorganic materials. This benefits some plants, but it is harmful to others, especially in winter when fluctuating soil temperatures can damage roots.
3. Light-colored rock mulches near buildings reflect heat in summer and may increase the need for air conditioning.
4. Rock mulch over black plastic is especially undesirable in planted areas because air and water infiltration is reduced. Soil under such cover often remains overly wet or dry.
5. Rock mulch placed directly on the soil surface eventually sinks into the soil. Removing it is usually back-breaking and tedious. Rock mulch is heavy and unpleasant to spread.
6. Leaves and other litter that collect in rock mulch are often difficult to remove.
7. Rock mulch may compact soil and does not improve soil structure or provide nutrients.
8. Rocks used as mulch may damage mowers and present other hazards when they edge or drift into mowed areas. They may be also be an attractive nuisance to kids.
On the plus side, rock mulch is relatively inexpensive, and it does not blow or float away or require renewal. Some consider it attractive. Others do not.
Although rock mulch does not attract rodents, neither do most organic mulches if they are properly applied and maintained. Rodents are attracted to a site primarily by food and shelter. It is the vegetation composition and structure, and not the mulch, that is the prime determinant here for rodent harborage.
As a side note, it is interesting that you are in the position of trying to locate information on the rodent-mulch connection. Challenge the homeowner’s association to show evidence that there IS a connection. They will come up short.