Shimano makes the TZ510 7-speed freewheel which is 14-34 teeth. It generally retails for under $20. You'll need to get a "freewheel removal tool" which is a splined tool on one side while the other side has a 1 inch hex nut. From Park Tool, the model FR 1.3, are less than $10 and a well equipped bike shop generally has these in stock. Be aware that a very similar looking tool is for cassette removal and doesn't work well on freewheels. The freewheel removal tool is longer overall than a cassette tool.
The increase in tooth count of the TZ510 will lower your gear ratios quite a bit. When I first experienced a rear end tooth count above 32 teeth, I thought for sure I could peddle right up a wall. Hills still require a good amount of rider input no matter how low a ratio one has, however they are a tremendous way to gain strength and endurance, especially so for a growing youth.
Back on the maintenance side of this, note that removing a freewheel oftentimes takes a very large amount of torque as the freewheel tends to tighten itself as the bike is ridden. I use a 1" box wrench that is about 18" long. A crescent wrench ( a brand of adjustable wrench) could be used as well, but my point is, the longer the better because it's gonna be tight.