you may measure the viscosity of a polyelectrolyte solution. First measure it for a PAA solution without added hydroxide (PAA in the protonated form). Afterwards add the hydroxide in measure the viscosity again. If the polyelectrolyte is a weak one, addition of hydroxide will lead to chain expansion and consequently to an increase in viscosity (of course you have to add sufficient amount of hydroxide in order to achieve some appreciable chain expansion). If the polyelectrolyte is a strong one (completely ionized (i.e. chains are already in an extended conformation) and addition of hydroxide will not effect the viscosity appreciably. The viscosity can be measured in a capilary viscosimeter.
Another option is to measure the conductivity. Weak polyelectrolytes with the same concentration as strong polyelectrolytes conduct the electric current much more weakly.
For very precise conductivity measurements you need a cell for measuring resistance of electrolyte solutions (two platinum electrodes with defined surface area S and distance l) and a Wheatston bridge for measuring the resistance of the cell. These measurements can be rather tedious. If you do not have a Wheatston bridge, you may measure the conductivity of the solution containing cell simply by a pH meter. These instruments often measure also conductance.
I would go for viscosity, if you have a capillary viscosimeter and a temperature bath to keep T constant.
You should measure the variation of the surface charge or the zeta potential of the polyelectrolyte, with the pH of the aqueous phase. No change should occur if the polyelectrolyte is strong. However, if the polyelectrolyte is weak, enhanced surface charge and/or zeta potential should occur as resulting from the deprotonation or the ionization of the polymer, due to the increase of the aqueous phase pH.
For more details see:
A. Jada et al. Surface charges of anionic and cationic polyelectrolytes and their effects on the nucleation and the growth of CaCO3 crystals. In e-Polymers 2009, no. 136. http://www.e-polymers.org.
Classical examples are poly(acrylic acid) (PAA, weak) and poly(styrenesulfonate, PSS) (strong):
In the plot of specific conductivity (SC) versus degree of neutralization (DN) for PSS decreases linearly from DN=0 up to DN=1 and finally increases for DN>1.
For PAA in contrast SC increases monotonously from DN=0 up to DN=1 and further for DN>1.