τὰ παιδία δώροις ἐλύοντο. τὰ παιδία δώροις ἐλύετο.
my answer I am sure is wrong…
The children were untying themselves their gifts. He was untying himself gifts for the children.
…Help Please. 
I would take the sentences as having an implied subject (they in ἐλύοντο; he/she in ἐλύετο)
with direct object τὰ παιδία and dative of means δώροις. The usage of middle-passive ending
is due to some relational connection to the subject, perhaps his/her in ἐλύοντο and their in ἐλύετο.
But isn’t the subject τα παιδια? The verb is in the passive. I think this rather illustrates the point that a neuter plural subject often takes a singular verb. So both sentences mean the same, “the children were bought free with gifts”.
Or rather, I think both interpretations, medium and passive, are possible depending on the context.
That’s certainly a possibility. In which case, only the second sentence with sg. verb is
grammatically correct.
Could it be something on the lines of
“The slaves ransomed themselves by means of gifts. He ransomed the slaves by means of gifts.”?
What is the context?
Not necessarily. See Smyth §959.
“A plural verb may be used when stress is laid on the fact that the neuter plural subject is composed of persons or of several parts”
I stand corrected, although I don’t quite understand what is the emphasis with a plural verb.