There are many ways this could be done.
Have them measure the thickness of the textbook and calculate from that the thickness of a single page. Averaging 20 or 30 of these measurements will give you a very accurate estimate. Measure the "real" thickness with a micrometer.
Give each student something round--coin, jar lid, etc.--and have them measure the diameter and the circumference with a ruler, or by wraping a string around it, or by rolling it along the table top. Perhaps have each student or pair of students use each of these three methods. Calculate the value of pi from the student data. You'll be surprised how many sig. digits you get by averaging a bunch of these together. Calculate mean and standard deviation by groups, such as by measurement method, or by groups of students.
Lay sheets of paper on the ground in front of all four tires of a car. Pull the car forward so each tire is in the center of a sheet. Use a pencil to trace the footprint of each tire onto it's sheet of paper. Do this carefully so the area of the tracing very closely matches the area footprint. Measure the area of each footprint and add them together. Measure the air pressure in the tires. The weight of the car is the total area of footprint multiplied by the average tire pressure. (or simplify it by making each tire have the same pressure before you begin). e.g. Weight = cm^2 x (kg / cm^2) or weight = in^2 x (lb / in^2) (I'm not sure what the proper metric units would be here, maybe Pascal?)
Make a rummy balance by balancing a meter stick in the middle on a knife edge. Place a coin of known weight at the 10 cm mark and move the fulcrum to balance the meter stick. Put a drop or a few drops of water at the 99 cm mark. Slide the coin toward the end to rebalance the meter stick. Do some arithmetic to calculate the weight of the water drop(s).