Coin cell CR2032

These really DO NOT like current spikes.  So a lot of buffer capacitance will be needed to buffer the LED current spike.  Data from Energizer:

• 235mAh downto 2V, rated for 190µA load
• Lithium / Manganese Dioxide (Li/MnO 2 )

Battery life calculations show that the project could run about 1 year when ON-current is 5mA and OFF-current is limited to 1µA.

3V CR123A Lithium cells

store up to 1700mAh.  These will last a lot longer.  Unfortunately they're much bigger : 34.5mm x ⌀17.0 mm

LIR2450

If a rechargeable solution is desired, then the 3.7V/120mAh LIR2450 can be used.  This cell tolerates current spikes much better.

The dimensions are acceptable as well 5.0mm x ⌀24.5 mm

Charger : Hackaday.io charger

Battery clip: MPD BK-880 & Keystone 3009