Thanks for the responses, they were helpful
Thanks for the relay part number 12193606. It happens that a relay with this part number was in the fuse box under the hood. I swapped the relays. The brake light relay with this part no. is only slightly less noisy. I wrapped the relay with some sound deadening material and foam and reduced the clicking sound a little more. I will continue to work on the issue. I guess I'm a little picky about unnecessary noises like loud clicking relays. Maybe there's a solid state relay out there that can be adapted to work for the brake light.