“…the meaning of התפלל is to make oneself the object of God’s פלל (assessment, intervention, or judging)”

Most likely, the hitpael form in the case of התפלל is doing the same thing as the hitpael form in the case of התחנן: it is turning the word into a voluntary passive/indirect reflexive. Hence, the meaning of התפלל is to make oneself the object of God’s פלל (assessment, intervention, or judging). This is a much simpler understanding of התפלל than the ones that look for a reflexive action on the petitioner’s part.

Mitchell First, “The Meaning of the Word Hitpallel (התפלל)”, The Seforim Blog (29 August 2016) [http://seforim.blogspot.com/2016/08/the-meaning-of-word-hitpallel.html]