Measure the volume of water in the container before putting the samples in, and again after taking them out

Yes, this would work. As the samples soak up liquid, the volume left in the container will become less. If you know how much water was there at the start, and measure how much is left at the end, you can work out how much was soaked up.

You could also weight the samples before and after putting them in the water.

How would you make sure it's a fair test?