create delay before the next click on doorbell
*** Important Note: this answer is not an endorsement for a particular manufacturer or distributor, you should use a search engine of your choice to look for &quot;100mA PTC&quot; , download datasheets, and make your own informed choices ***
Assuming you have a standard doorbell , that operates off a bell transformer (up in the roof) with 12VAC fed to the switch , and a solenoid type ringer that has two metal chimes (ding when pressed, dong when released) , then the simplest option would be to install a PTC (positive temperature coefficient) thermistor or poly-fuse ,  these are normally designed to protect low voltage circuits instead of fuses, as they will reset once the fault clears. The thermistor heats up when excessive current passes, and the resistance increase dramatically. PTC's are commonly sold in electronic hobby shops as &quot;speaker protectors&quot; , they are usually yellow discs about 10mm diameter.  
In your case, I'd guess your coil is about 1amp at 12v, so I would be looking at possibly a 100mA PTC  to get a 1second delay see here https://au.mouser.com/Circuit-Protection/Thermistors/Resettable-Fuses-PPTC/_/N-axfyt?P=1z0z7l5&amp;Keyword=PTC+100mA&amp;FS=True  . the first part in that search has a datasheet , https://au.mouser.com/datasheet/2/643/ds-CP-0ZRR-series-1664153.pdf , please read this datasheet, from this we get a time delay graph :  , I have highlighted in red the region of interest , a 100mA PTC is curve B,  if you look along the X axis to 1amp (your bell current) and then go up to curve B , it meets at about 0.5sec, so the bell coil will release in about 0.5sec.
In order to get the optimum time delay, I'd suggest buying about 4 of the 50mA PTC's and wire up as many as needed in parallel to get the required delay. The PTC's go in series with the pushbutton, and are quite small, so will fit inside the button assembly, simply remove the pushbutton cover, disconnect one wire, and connect the PTC between that wire and the screw terminal, use a wire nut to join the other PTC lead to the wire.



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