acceptable bounds for replacement DC adapter
If the old supply put out 8 V and 2.6 A and the equipment says it wants 8 V, then the obvious answer is a power supply that puts out 8 V and 2.6 or more amps.  The equipment might be OK with 9 V, or not.  It could operate mostly fine, but in some corner case overheat, suddenly stop operating, or vanish into a greasy black mushroom cloud.

You are right in that 8 V is not a standard value.  However, many OEM power supplies can be tweaked a few 10s of percent from their nominal value with a trimpot or something.  Manufacturers know that lots of different voltages are required but can't stock a different supply for every weird voltage a customer wants.  Take a careful look at data sheets of some "9 V" OEM supplies, and you will probably find most of them can be convinced to do 8 V.

I just wrote a detailed answer about this general topic.