Acts 20:28 is sometimes rendered - “... to shepherd [“feed” in some translations] the church of God which He purchased with His own blood.” - NASB
But there are 2 major uncertainties about this translation of Acts 20:28.
First, even some Bibles translate this verse, “the church of the Lord.” - NEB; REB; ASV; Moffatt. "to feed the church of the Lord which he purchased with his own blood." - ASV.
The popular The New International Dictionary of New Testament Theology, p. 838, Vol. 2, Zondervan Publ., 1986, also uses this translation for Acts 20:28: “to feed the church of the Lord”!
And the respected, scholarly work, A Textual Commentary on the Greek New Testament, p. 480, United Bible Societies, 1971, explains about this first uncertainty concerning the translation of Acts 20:28. It admits that there is “considerable degree of doubt” about the two different renderings. It states that “The external evidence is singularly balanced between ‘church of God’ and ‘church of the Lord.’”
Second, some Bibles render this verse, “to care for the church of God which he obtained with the blood of his own Son.” - RSV, 1971 ed.; NRSV; NJB; CEB; CJB; GNT; LEB; MOUNCE; NCV; NET).
The New Testament Greek words tou idiou follow “through the blood” in this scripture. This could be translated as “through [by means of] the blood of his own.” A singular noun may be understood to follow “his own.” This could be referring to God’s “closest relation,” his only-begotten Son.
Noted scholar J. H. Moulton says about this:
“something should be said about the use of [ho idios, which includes tou idiou] without a noun expressed. This occurs in Jn 1:11, 13:1; Ac 4:23, 24:23. In the papyri we find the singular used thus as a term of endearment to near relations .... In Expos. vi. iii. 277 I ventured to cite this as a possible encouragement to those (including B. Weiss) who would translate Acts 20:28 ‘the blood of one who was his own.’” - A Grammar of New Testament Greek, Vol. 1 (Prolegomena), 1930 ed., p. 90.
New Testament scholars Westcott and Hort present an alternate reason for a similar rendering:
“it is by no means impossible that YIOY [huiou, or ‘of the Son’] dropped out [was inadvertently left out during copying] after TOYIDIOY [tou idiou, or ‘of his own’] at some very early transcription affecting all existing documents. Its insertion [restoration] leaves the whole passage free from difficulty of any kind.” - The New Testament in the Original Greek, Vol. 2, pp. 99, 100 of the Appendix.
And A Textual Commentary on the Greek New Testament, p. 481, tells us:
“Instead of the usual meaning of dia tou haimatos tou idiou [‘through the blood of the own’], it is possible that the writer of Acts intended his readers to understand the expression to mean ‘with the blood of his Own.’ (It is not necessary to suppose, with Hort, that huiou may have dropped out after tou idiou, though palaeographically such an omission would have been easy.) This absolute use of ho idios is found in Greek papyri as a term of endearment referring to near relatives. It is possible, therefore, that ‘his Own’ (ho idios) was a title which early Christians gave to Jesus, comparable to ‘the Beloved’.”
Therefore, we can see that a rendering similar to RSV’s “the church of God which he obtained with the blood of his own son [or ‘beloved’]” is an honest rendering.
Although the UBS Committee didn’t actually commit itself one way or another on this rendering of tou idiou at Acts 20:28, it did mention that “some have thought [it] to be a slight probability that tou idiou is used here as the equivalent of tou idiou huiou [‘his own Son’].” - p. 481. Obviously this includes those scholars who translated the Revised Standard Version (1971 ed.) and Today’s English Version.
The The NIV Study Bible states in a footnote for Acts 20:28:
"his own blood. Lit. 'the blood of his own one,' a term of endearment (such as 'his own dear one,' referring to his own Son)." - NIVSB, Zondervan, 1985.