Any compatible trigger shifters for el-cheapo TD ONE derailleur? Bought a cheap RoadMaster MTB's as a first "grown up" bikes for my daughter. For the price (~$60 in a sale) they play nicely, but it shows the poor quality of the components.

The biggest gripe is the grip shifters (as in each and every Walmart bikes). I'd like to change the shifters to trigger type. The rear cassette is  6-speed and the derailleur (as the rest of components) is some "TD ONE" generic brand. I know the shifter must match the actual derailleur type, due to differences in the pull ratios between brands.

Any suggestion on what type of shifter would be compatible with the TD ONE derailleur?
You need a 6 speed shifter.  These are likely to be rare, so a 7 or 8 speed shifter should work, but will have 1 or 2 ghost positions which may be confusing.

Another possibility is to use a friction shifter, where the rider has analogue input and can put the chain where it needs to go.   

But from another point of view, you're throwing good money after bad.  A cheap bike is okay while they're still growing, but at some point you're better off saving for a good bike.

If there's a bike cooperative in your area then they may be able to assist with take-off parts or similar.