Does a homeowners insurance cover a tenant getting bitten by a brown recluse spider inside the home?

In general No. Insect bites are usually in the arena of a health insurance policy, not a property insurance policy.

A property owner is not automatically liable for injury to a person simply because the injury or insect bite occurred on your property. To establish financial liability the injured person would have to show that the property owner was the direct or proximate cause of the sustained injury, either through action or inaction contrary to that which a prudent person would take.

If you feel you may be liable for the insect bite, You should first check your policy and establish whether or not you purchased personal Liability coverage with your policy. If Liability coverage is in place. Report it to your company. Your insuance company will provide coverage for the cost of legal defense for you if necessary and would pay for any damage awarded against you up to your liability limits.

Property insurance should never be construed to take the place of a health insurance policy.