Beryllium copper is commonly use for springs in electrical contacts. It is fairly low resistance, like copper, but unlike copper, it is springy when properly heat treated, and can withstand a near infinite number of flex relax cycles.
...But...
It is only occasionally used, by itself, as a contact material. It presents no special characteristics to that task. More commonly, a more suitable contact material is fused, or riveted to the beryllium copper spring, to form the contact assembly of a relay, or solenoid.
When it is talked of beryllium copper being non-sparking, what is being discussed is the lack of a glowing hot metal particle being dislodged when the metal is rapidly mechanically cut.... unlike steels. This property makes beryllium copper a valuable material for making hammers, chisels, and other tools used in explosive atmospheres... like around a leaking gasoline pipeline.
I am certain that most cheap starter solenoids are using a nearly pure, but work hardened, copper alloy for their contacts, Premium starter solenoids use a copper nickel silver alloy for their contacts because of its low welding and material transfer characteristics.
If you do a quick google search, you will find that virtually all starter solenoid parts, needed to rebuild your solenoid, are readily available.
-Chuck