There is a big difference between a "additional named insured" and an "additional insured".
One little word (named) makes a world of difference in the coverage that is afforded under the policy. A additional named insured has all of the same coverage afforded the first named insured and in fact can exhaust the limits of the policy without the first named insured having any connection to the loss at all. Example - BO's horse gets loose, runs in the road, is hit by a car and the injured in the vehicle sue BO for injuries. BO files claim under this policy, after all they are a named insured and the policy pays out the $1,000,000 limit on the policy, Policy limits are now exhausted (through no fault of First named insured) so if first named insured (OP in this case) has an incident resulting in a claim there are no funds under the policy to pay the claim.
Now on to "additional insured". In this case the BO would ONLY be afforded coverage under the policy IF the named insureds (OP) horse caused injury (kicked a kid in the head at a show for example) and suit was filed against the named insured and the BO (yes the BO would probably be dragged into this even if it was an offsite show because in today's sue happy society everyone is looking for the deepest pockets).
In the OP's case I would be willing to bet the BO is looking to be added as an "additional insured" rather than an "additional named insured" and if that is not the case, run and run fast. I can think of any insurance carrier that would be willing to add an additional named insured endorsement on an individual horse owners liability policy adding the BO because they would be taking on the exposure for everything that happens on the property.
bluemooncowgirl - a umbrella liability insurance policy is an additional layer of coverage that requires primary underlying coverages (i.e. personal liability on your homeowners policy and/or your auto liability on your auto policy) at specified minimum limits before it will respond to a loss. Example - you are sued for $1,500,000 - your primary/underlying limit on your homeowners is $1,000,000 but you have a $1,000,000 umbrella. Between the 2 policies you would have sufficient limits to cover the judgment if the original claim was covered under the primary personal liability.
And before anyone asks, yes I have been an agent for 24 years now.