Weak Precedence Grammar and Parsing
Weak precedence is first an algorithm for parsing (see Parsing Techniques - A Practical Guide 1st edition p. 186). So a weak precedence grammar would be a grammar which can be parsed by this algorithm (see the reference for a description of the algorithm, you'll probably want to start at least a few pages earlier).

IIRC, the interest of weak precedence parsing dropped at lot when practical LR(1) variants (SLR(1) and LALR(1)) were discovered, but precedence algorithms are still sometimes used in hand written parser for parsing expressions while the rest of the grammar is parsed with a recursive descend variant.