Concept:

Turn an adult IKEA stool into a 12-15" adjustable-height child's cello/music stool
The end result is extremely stable and comfortable for the child
Add a handle to the bottom for easy carrying

Overview:

You're going to drill a series of holes along each leg, then cut the legs in half and bolt back together through the holes
Use scrap wood to build a simple drilling guide so that each leg ends up being identical
Total cost approximately $30 (including stool)
Total time approximately one hour

Alternative:

Given that the stool only costs $20, you could also just buy a few of them and saw off the legs at different heights. But that wouldn't be as fun or functional.

Step 1: Gather Materials

Supplies needed:

1x Ikea Frosta stool
8x 1/4" x 2.25" or 2.5" bolts (shorter is better, but not a big deal)
8x 1/4" wingnuts
16x 1/4" washers
1x handle (door pull type)
Scrap wood to build a drill guide (8-12"-ish square of plywood, two wood blocks as thick or slightly thicker than the legs, two more small boards)
Nails or screws to build the drill guide

Tools needed:

Step 2: Mark Up the Legs for Drilling and Cutting

Line up the legs so that they are even across the feet (top of the legs curving up)
Number the top and bottom of each leg so that they can stay matched up after cutting
Draw two lines across all four legs:
1. One at 1.5" from the bottom
2. One at 7.5"
On leg #1 add a third line at 2.5" from the bottom
Turn leg #1 onto its side (either side) and add a fourth line at 7"

Place leg #1 on the scrap of plywood
Screw down a piece of scrap wood on either side of the leg so that it now sits firmly (but not stuck) in a channel
Prepare the spanning block: take another scrap piece of wood that can sit perpendicularly across the leg and drill a perfectly straight 1/4" hole 1/2" or less from the block's edge. Put it aside for now
Clamp down the drill guide and leg extension so that the drill press is perfectly centered on the 1.5" pencil mark
Drill a 1/4" hole centered perfectly on the lower line
Without moving the guide, slide the leg up the channel so that the drill is now centered over the 2.5" pencil mark
Drop a 1/4" bolt through the spanning block and into the first hole you just cut
Screw the spanning block firmly onto the guide blocks
You now have a consistent 1" spacer for all future holes

Make sure the spacing bolt is firmly in the first hole you drilled
Drill the second hole, which should now be exactly 1" from the first
Slide the leg up and drop the spacing bolt into the second hole
Repeat until you have five evenly-spaced holes
Move the leg up to the 7.5" line and drill an additional five evenly-spaced holes
Repeat for all four legs
You should now have drilled a total of 40 holes, 10 on each legs in two sets of five

Notes:

It is extremely difficult to do this with a hand drill (built the first stool using a hand drill - it was messy and difficult but did get it to work)
Note in the photo that the holes aren't perfectly aligned - there was a bit of play on the guide block that I fixed mid-process by really screwing it down. Note that the pieces still fit together, but it's a tighter fit

Set mitre saw to 45 degrees
Line up leg #1 with the blade sitting at the 7" line on the side of the leg
Make sure the angled cut starts on the outside of the leg at the pencil mark and angles towards the foot so that the inside of the leg ends up longer than the outside
Clamp down the leg
Clamp a scrap block of wood firmly against the end of the leg
Cut all four legs, using the scrap block as a length guide
You should now have four identical curved legs and four identical leg extensions

Bolt Legs Together:

Match up each curved leg with its extension
Line them up so that the extension sits inside the curved leg with the long edges of the leg and extension touching, and the holes aligned
Insert 1/4" bolts through the top and bottom hole of each leg pair with a washer on either side and secured on the inside with a wingnut
You may need a hammer to make the parts fit together

Assemble the Stool:

Use the hex key and supplied hardware to bolt the legs onto the stool
Screw the handle onto the bottom of the stool, in between two legs and lined up with the stool's edge

Adjust the Stool:

Sand the bottom of one more legs as needed to make the stool level
The stool will sit approx. 12" high when all five holes are aligned
Shift the leg extension down and bolt through the top and bottom available holes to adjust in 1" increments up to 15"