What is the name of an attack where the compromise of one key compromises the whole system? Consider the following: A whole system depends on the security of one private key and if this key is, for any reason, compromised, the whole system will be compromised.

What's a suitable name for such an attack?
The whole system depends on the security of one private key (..that gets..) compromised.
  Give a suitable name for such an attack.


I would call this class of attacks: Master key compromise.



Examples of such attacks (as asked in comment) include


The master key is revealed to an attacker by physically stealing a backup copy, bribery or duress (obligatory XKCD) of a holder of the master key. Mitigations use safes, and/or k-out-of-m secret sharing.
The adversary extracts the master key from a device holding it (the most obvious example of that is when the master key is in every device, which is not unseen). Mitigations include using security-certified device at least for master keys (HSMs, SAMs, Smart Cards). Method of extraction include


directly reading the memory/medium that stores the key,
examining memory where the device using the key copies it, perhaps temporarily (e.g. with a debugger, a JTAG port, exploitation of a buffer overflow vulnerability somewhere in the code..)
timing attack, (differential) power analysis and similar side-channel attacks (including fault attacks)

Cryptanalytic attack or/and brute force attack, where a secret key (including private) is inferred from public material, like ciphertext, plaintext made available to an adversary, public key. Mitigation include using larger security parameters for master keys.