How 'Avalanche Effect' got its name?
The term first described by Horst Feistel (1973)

As the input moves through successive layers the pattern of 1's generated is amplified and results in an unpredictable avalanche.

It is rather very good naming. Consider a bit flip in the AES's input; It enters the first Sbox, then according to the design of the Sboxes of AES the output is different since they are invertible (actually they are also designed to have avalanche effect). Now say we have at least two different output bits, then the Sub-Bytes and MixColums will distribute the changes. Therefore in the first round, you will have 2 changes, in the next round you will have 4,.. and so on. As you can see the minimum change becomes larger and larger in each round. That is how an avalanche is working.
Note that we have $2^{10}$ changes in the last round for AES-128. Contradictory, no, since some changes will cancel each other and that part together with the key's action is unpredictable for our computing capabilities!
The blow picture was drawn by Feistel, which should tell the idea; One small change at the beginning and it is a big effect on the output.