I believe it is to indicate they understand this is a Messianic reference to Jesus Christ. Peter preaches most of this Psalm and emphasizes this verse in Acts 2 to prove by Christ's resurrection from the dead, which they were all witnesses to, therefore that Jesus must be the Messiah (Christ, "Anointed One") of Psalm 16:10. Here's my sermon on this (first for a seminary class of OT quoted in the NT, here for our church when I first arrived):
https://www.sermonaudio.com/sermoninfo.asp?SID=92010013339. Specifically related to the last question, Peter says David knew he was a prophet speaking of Christ's resurrection in this verse according to Acts 2:29-31.
While the ESV and old Geneva have it lower case, by a quick perusal I see that the NAS and the Almeida Portuguese ("Santo" -- Spanish RVR has it lowercase) have it capitalized. I notice that all I reference here have it capitalized when quoting it in the NT in Acts 2:27 and 13:35 (though Tyndale doesn't, written simply as "holye"). Both verses in the Greek do not have it capitalized (neither in Psalm 16:10 in the OT, though there really isn't capitalization in Hebrew as its all essentially upper case the way I see it). Interestingly, in Psalm 2:2, "anointed" which is clearly also referring to the Messiah is lower case in the KJV (not the Portuguese and Spanish), but upper case in the rest I'm referencing including the ESV actually (old Geneva even translates it as "Christ"). Similarly, Psalm 2:7 has "Son" capitalized as it is in Acts 13:33 especially relevant related to the 13:35 reference above, including the old Geneva ("Sonne": Tyndale doesn't translate); the Portuguese also has it upper case ("Filho") but the Spanish lower ("hijo"). So while all recognize these as Messianic, they have variety even within the translations, on whether they capitalize each one. Easy for me to Monday morning quarterback here, but I think I would have made all references I note capitalized for consistency and quick reference.