What battery should I use in a remote control for a score marker?
The battery choice depends on how you intend to implement the wireless communication between the control unit and the score marker.
There are several options you can opt for, but you are mostly limited by the range you require.

It is clear that you will only need to transmit to the score market only when you press a button, therefore the controller can go into a sleep or deep sleep mode in the meanwhile, extending quite considerably the battery life.
Here are a few options:


The simplest solution is ag ate opener remote running at 433MHz which usually have 4 buttons. These remotes usually run on a CR2032, which will last years, It is extremely cheap, you can potentially hack one of these if 4 buttons are enough.
NRF24L01 - it uses 2.4GHz frequency, and it can be quite good low power mode but you will need another MCU to manage the buttons, data to transmit, and power management. Even with that, a good AA battery would work for a good part of a year, as long as you manage smartly the sleep modes.
The  datasheet of the NRF24L01 states at page 21 that the current consumption is 900 nA in Power Down more and 26 uA in Standby. If you use an Arduino Pro mini, with the appropriate modifications (such as removing the LED), you can take the current consumption as well in the uA range.
I would avoid WiFi and Bluetooth, as they introduce a lot of unnecessary complications in a project that requires a low bandwidth and some range.


If you are keen on using rechargeable batteries, then you can use them, but I would stay away from 18650 because they are rather bulky for a remote. I would rather go for smaller ones in a rectangular form factor, such as the ones they use for flying small drones, which have a capacity of 500-1000mAh. These batteries are still overkill, but can work, as long as you choose a type with a low self discharge rate.