Shimano Claris cassette upgrade

11-32 will definitely work - search for an HG-41, usually under £20. You'll need a new chain too - about £15. Chains are speed-specific, so you need an 8sp/9sp chain.

Tools wise you'll need a chain whip and cassette tool and chain tool. It's a relatively easy job as tolerances on 8sp are such that you probably won't need to reindex the gears.

Steps:

1. Break old chain using chain tool

2. Remove rear wheel

3. Remove cassette using cassette tool and chain whip. The cassette locking is usually done up bastard tight but take plenty of abuse/strong words. Think car wheelnut tight.

4. Clean and lightly grease the freehub body once the cassette is off

5. Slide new cassette on - it's keyed so will only go on one way

6. Tighten cassette lockring - you'll hear it make grinding noises as you do this, don't worry. Torque setting will be on the lockring. Tighter the better (within tolerances) as it will prevent freehub wear

7. Install rear wheel.

8. Size chain. Place it on the big ring at the front and rear but don't run it through the derailleur. Plenty of guides online on how to do this, as always Calvin from Park Tools is your man. 

9. Once sized thread the chain through the derailleur. It does round the front of the top pulley and round the back of the bottom pulley. Make sure you also thread it rearward of the chain retention tab on the jockey cage halfway down between the pulleys.

10. Install chain pin using chain tool, or quicklink if you prefer. There's a bit of an art to getting the chain pin right, quicklinks are more foolproof.

11. All done, check it shifts OK, particularly when the chain is on the big ring at the back as the derailleur B-screw might need winding in to give enough clearance. If the indexing is off all it will take is small moves on the barrel adjuster on the rear derailleur. Quarter or at most a half turn at a time. 

According to Shimano the CLARIS Short Cage 8-speed Rear Derailleur has a low sprocket max of 32t and the medium cage will take 34t but those are the latest models so yours could be different. 

https://bike.shimano.com/en-EU/product/component/claris-r2000/RD-R2000-S...

https://bike.shimano.com/en-EU/product/component/claris-r2000/RD-R2000-G...

I also found this to work out the total drivetrain capacity:

Subtract the number of teeth on the smallest chainring from the number of teeth on the largest chainring.
 50-34= 16.

Subtract the number of teeth on the smallest cassette cog from the number of teeth on the biggest cassette cog
 32-11=21.

Now add the result of step 1 (crankset capacity) to the result of step 2 (derailleur capacity).
 16+21=37.

You end up with a total drivetrain capacity of 37 for a 50/34 crankset with a 11/32 cassette.

Shimano says the max capacity is 37t for the short cage and 43t for the medium but If your Mum's crank has different size sprockets then you'll have to do the sums yourself.

If you're fitting a new chain it looks like a 11-32t will just fit if she has a short cage and comfortably fit a medium cage but if you aren't 100% sure then i'd go for a 11-28 to be on the safe side.

On another point i've been unlucky with cheap Shimano cassettes from ebay and amazon in the last couple of years to the point where I became convinced they were either fakes or Shimano QC has gone downhill. Now I only buy HG41 or better from online bike stores.

Found this on a review website. They seem to think it would be OK.

"Because Shimano Claris is an eight-speed system (i.e. there are eight sprockets on the cassette) there are always going to be bigger jumps between gears, compared to the nine-speed Shimano Sora, 10-speed Shimano Tiagra, and 11-speed 105, Ultegra, and Dura-Ace groupsets. This potential problem is compounded by the fact that Claris comes as standard with a medium cage derailleur, which means that you can fit a 32-tooth rear sprocket, spreading the gear range wider."

I put 11-32 on my Sora and it works fine.