I am not expert, and maybe some of my contributions are not new, but just wanted to contribute. In addition to suggestions of others, I would think of two different buckets:
1. Businesses to leverage the capabilities of an insurance company:
- Data analysis: insurance companies record a lot of data (e.g. for a car insurance, car model, driver age, location, driven miles per year, frequency of accidents) that could be sell to third companies.
- Private investment fund: insurance companies have expertise on investing.
2. Businesses to reduce weaknesses of the insurance company (basically to reduce claims/costs):
- Private security
- Maintenance services
- Repair services