The TZ31 should be fine. Stuff to look for to make your decision:
1) All of the six, seven, and eight speed Shimano rear derailleurs use the same pull ratio so, as you suspected, all will work as far as the indexed shifting goes.
2) You want a derailleur capacity adequate for the bike. This is basically how much chain the derailleur can take up in moving from the gear combination that uses the most chain to the combination that uses the least.
3) You want a derailleur that can clear your largest cog at the rear wheel of your bike.
#2 and #3 will be listed in the product specifications for the rear derailleurs. As such, you can ensure that you choose an appropriate derailleur one of two ways:
a) Ensure that the specs of the replacement rear derailleur suit the gearing on your daughter's bike or;
b) Ensure that the specs of the new derailleur match or exceed those of the original derailleur.