hi William
Technically, there is more than one way to harmonize a melody (harmonize = to put chords underneath the melody). In the case of Twinkle Twinkle Little Star, one could harmonize the entire song easily with I, IV en V chord. Important is that you know what these chords mean and what notes they contain whatever key you choose to play the song in. To begin with, try to harmonize the notes that are stressed (usually on or around beat 1 and 3 in the measure). To harmonize the stressed note should be part of your chord (either 1, 3 or 5 of the chord). This is a good place to start, it's not the only way, because one could also choose to make the stressed note the 7th (major or minor) or the 6th of the underlying chord. This would give different results. That's why sometimes reharmonized songs can sound so different (check out Jacob Collier cover songs of "Moon River", "Fix You" or "Every Little Thing She Does is Magic". I think it helps to take a thorough look at the harmony sections in here to help you along the way as well. In the end for the choice of the chords it's important to not just rely on theory only but learn to trust your ears (ear training indeed).