Simple solution to send a short pulse to Solenoid for milliseconds?
simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. The low-tech solution.

How it works:


C1 is charged via R1 while the SW1 is open.
On pressing SW1 a high current flows from the capacitor to the solenoid which pulls in giving the "DING".
The capacitor will discharge and, with R1 chosen correctly, there will be enough current to hold the solenoid weakly in the extended position. (Remember that it will be over-driven to accelerate it hard enough to reach the chime with some velocity.) The idea here is that the holding current is reduced to a safe long-term level.
On releasing the button the solenoid will retract giving a "DONG".


Time for some measurements and maths:


Measure the solenoid current. You can do this by connecting a multimeter in amps mode across the button. 
Work out the coil resistance from \$ R = \frac  {V}{I} = \frac {12}{I} \$ or measure it.
You now need to figure out how much resistance you need to hold the solenoid on adequately. I would guess that a car indicator or brake-lamp bulb would be adequate for R1. You can test by wiring it across SW1, pressing SW1, listen for the "DING" and release. The lamp should be dim and the solenoid will be weakly extended if we've got this right.
Now for C1: Assuming about 1 A for the solenoid the resistance would be about 12 Ω. If we aim for a 100 ms pulse time we can calculate C from the time constant formula \$ \tau = RC \$. Rearranging we get \$ C = \frac {\tau}{R} = \frac {0.1}{12} = 0.01 \ \mathrm F \$ approx. This is 10 mF or 10,000 µF. You need 16 V caps (the next standard voltage).


This isn't quite what you asked for in the comments but retains the normal "DING-DONG" for a short press and gives a "DING" on press and "DONG" on release for the longer press.

Let us know if this works.