What is "hard quantization" strategy?
Suppose you are given a data point $D$ and want to classify it into one of $N$ possible classes, $C_i$ for $i=1,2,\ldots, N$.

Hard classification chooses one of the $C_i$ to represent the classification of $D$.

Soft classification can be done in a number of ways, but the usual (for me) is to generate a weight $w_i$ for each of the $C_i$ which indicates the probability that the data point $D$ is part of the class $C_i$. This means that 
$$\sum_{i=1}^N w_i = 1$$.

I haven't heard the idea associated with quantization of one dimensional signals, but I've seen it used in relation to vector quantization which can be thought of as a simple classification technique.



This paper seems to have an example that matches what you say in your comment.

The idea is that, instead of each code word in the dictionary having the same weight, each code word has a weight dependent on some criterion or criteria: how far it is from other code words; how frequently it is likely to appear, etc. etc.

The picture below gives two examples (green and light blue) of data points that are better coded using a soft technique than a hard one.