If my opponent uses Mime Jr's "Mimed Games" attack and they don't have enough cards in the Lost Zone, can I choose Sableye's "Lost Mine" attack and if so what happens?
No, you may not choose that attack, you must select a different attack.
Source: TPCi Rules Team (2024-02-01)

Relevant cards

Note: All applicable cards may not be listed below.
SV45_31
SV45_31
SWSH11_70
SWSH11_70