A lamp caught fire, damaging the outlet and causing $400 damage to the flooring. The electrician says the circuit is fine so the tenant's lamp must have been faulty. 

Any advice on whether I should just pay for repairs, try to get the tenant to pay for it, or claim it on my insurance?

Thanks!

You would treat it as any other tenant damage. During the term of the lease any damage you find you repair and bill the tenant.

If they ask you should tell them that the cost is never deducted from their deposit.

If they are smart tenants they will have tenant insurance coverage.