Security of AES ECB for 128 bit random data
AES is designed so that it is infeasible to derive the key with only the plaintext and ciphertext. It's part of the definition of a strong cipher.  

At the risk of sounding like a broken record, the real problem with ECB is that the same input blocks produce the same output blocks. This is devastating if your data is more than 128 bits. Other modes prevent this by adding in some randomness as well as having previous blocks affecting future blocks.  

Another security feature added by some modes is authentication. They provide a way to prove your data came from the place you expect it did, and to prove it wasn't tampered with. Without authentication, attackers could corrupt a single block of your data without you being able to directly tell (barring non cryptographic methods, such as checking if the decrypted data makes sense, but thats a lot less strong).  

If all you need is secrecy, and integrity and authentication is of absolutely no concern to your threat model, and you guarantee the same block isn't ever encrypted twice, then encrypting 16 bytes (128 bits) of random data with AES ECB is okay.