The pearson correlation, when squared, gives the % of variance (in the strict mathematical sense) that is explained by this variable alone.
So how much is 'weak' or not depends on the situation. If you have something that is influenced by multiple causes, then it is impossible to have any single high correlation. But any correlation, if statistically significant (i.e. highly unlikely to be caused by chance) can be important.
The question is, how much is an 'interesting' amount of the total variance, in the given situation.
(In the topic under discussion here, I have no idea.)