Testing for weak instrument
Staiger and Stock (1997) formalized the definition of “weak
  instruments” and most researchers seem to have concluded (incorrectly)
  from that work (or hearsay) that if the F-statistic on the excluded
  instruments in the first stage is greater than 10, one need worry no
  further about weak instruments.Stock and Yogo (2005) go into more
  detail and provide useful rules of thumb regarding the weakness of
  instruments based on a statistic due to Cragg and Donald (1993).
  Stock, Wright, and Yogo (2002) provide a summary of this work.


Weak instruments:  An overview and new techniques

Austin Nichols (2006)

While an  F-statistic cutoff of 10 may be a crude rule of thumb and not sufficient to conclude that a set of instruments are not weak, Stock, Wright, and Yogo (2002) have a table about weak instrument cutoff values that goes down to 1 instrument.

GMM, Weak Instruments, and Weak Identification