DO THIS!!!!
Grandma will adore it now and treasure it always... someday, it will probably come back to you. I have one of these that came from Hallmark pre-decorated and my girls take turns each year opening the doors (one gets "odds", the other gets "evens")
AC Moore has a brand of craft paints that is only $.59 a bottle and frequently goes down to $.33 on sale. They have packs of paint brushes for $5, and you can use the coupon (split your sale) to make that 40 or 50% off as well. If you use a 50% on the brushes and get green, silver, gold, and red paint (green tree, gold "garland", gold/silver/red blobs for the ball ornaments), you'll be out $20, total for the whole project.
Alternately, you can find a pretty Christmas picture, print it out 11X13, and decopage it over the whole thing. You'd need to take off the knobs from the drawers first, then put them back on through the paper after, but it would not be too hard. You might need some kind of trim to go around the edges if it's bigger than 9x13, or you could just leave a dark green border.
You didn't mention how old the kids are, but if they are young, that will be about as much art as they can handle. When they are older, they could offer to re-decorate the tree with more detail (don't be surprised if Grandma refuses, she'll love it however it already is...), or you could buy a pack of mini ornaments on clearance after christmas each year and add them on the next year.
As for what goes in the drawers, I know the piece you're talking about, and the drawers are about the size of a ring-box, so that is your limit, gift cards won't fit... but I *LOVED* the PP's suggestion of notes.
-Maybe each drawer has a note/picture from one of the girls (alternate on odd and even days) telling/showing something she loves about Grandma, or recalling a special moment of the past year.
-Over the coming year, you could collect trinkets from memorable moments for inclusion next year "Dear Grandma, here is an acorn from our nature walk, I will always remember it and hope you will too!"
- In the drawer you mark "24", hide a little thumb drive (they make them small enough) with all the pictures from the past year.
-One year, you could hide her special present somewhere, and write the clues to the location on a piece of paper, glue it to cardboard, and cut it into 24 puzzle pieces... she has to open all 24 drawers to get the pieces, then solve the puzzle to find her present... (you'll have to make sure the "giveaway" pieces are the last drawers to be opened.) (oh, you can also buy pre-cut puzzle cardboard blanks... but we're being budget-y, after all.)
Again, DO THIS!!! I wish these had been available before I spent $65 on my Hallmark one years ago.