The question is being asked because Jesus teaches us to pray to the Father (Matthew 6:9) in his name (John 14:13-14). It makes sense on an intuitive level that one would supplicate one Person (the Father) in the name of another (the Son). But to supplicate the Father in the name of the Father (in his own name?) and Son and Holy Ghost, or to supplicate the Father, Son and Holy Ghost all at once in the name of the Father, Son, and Holy Ghost, makes little logical sense. It sounds like those prayers people make in public when there are Deists around so they prayer to God and end with "in YOUR name" rather than "in Jesus' name." Supplicating someone in their own name doesn't make sense. Anyway, all of that aside, scripture only mentions praying to the Father in Jesus' name, so how can the custom of closing a prayer "in the name of the Father, Son, and Holy Ghost" be justified scripturally? Or do the Protestant traditions that engage in this practice simply drop Sola Scriptura on this practice and justify it by Catholic tradition?
Asking why "some protestants" do close their prayers in a particular manner, while seemingly limited, is still quite a broad question, in fact so broad as to probably be unanswerable. I submit that it would be more profitable to ask a particular protestant why he or she ended a particular prayer with the particular formula you cite. I rather suspect that in the case of most extemporaneous prayer, most of the time the answer you will receive will be on the order of "it seemed the right thing to do at the time", or "the Spirit inspired me to do so". Further, as to justifying a particular close to a particular prayer, I can't think of any protestants I know who would feel a need to justify a particular form of closing prayer in a particular circumstance.
Can you provide some examples of prayers ended with the formula you cite? None of the prayers in the 1662 Book of Common Prayer of the Church of England, and other prayer books in my collection, I don't find any examples that end with the formula, "in the name of..." that you cite. Every one I examined end with some variation of "through Our Lord Jesus Christ...".