Here are the main options I have found:
- The author of Hebrews boldly and matter-of-fact-ly takes a clear reference to God, the LORD, and attributes it to Jesus. While it may seem like a misuse of Scripture, the argument goes that it is done on sound theological basis and logical syllogism: since Jesus is God, even LORD (= YHWH), therefore any reference to God or LORD in the Old Testament can be legitimately also transferred to Jesus. It doesn't matter at all whether there are any specific indicators for the quotation that it has reference to the Christ, because the basis of usage is not on the "Messiah-ship" of Jesus, but rather just to do with his "LORD-ship". It can also take its lead from the argument of Jesus in John 10:35-36. This is a position we see in the commentary by Philip Edgcumbe Hughes (pp.67-68).
- The author of Hebrews attributes to Jesus a reference to God because the reference also has Messianic indicators. This option is a bit more focused and nuanced than the first option, but includes the argument of the first option. Not only does the author feel free to use a reference to God on the basis of the divinity of Christ, but also because the Psalm is referring to God in ways that should also be considered Messianic or relating to the work and coming of the Christ. Calvin's commentary takes this approach. He says, "I indeed allow that Christ is not named in any part of the Psalm; but it is yet plain that he is so pointed out, that no one can doubt but that his kingdom is there avowedly recommended to us. Hence all the things which are found there, are to be applied to his person; for in none have they been fulfilled but in Christ...".
- The author of Hebrews understands the quotation to actually be words of God addressed to the Messiah. This option sees Psalm 102:25-27 as words spoken by God addressed to the Psalmist (or perhaps some other person), understood to be the Messiah. The beginning of the quotation of God's address starts in the middle of verse 23, according to the LXX translation, and this option understands the author of Hebrews as following the same reading as does the LXX. (The difference between the LXX and traditional Hebrew rendering is accounted for by the question of what Hebrew vowels to use in a certain word.) This option also finds strength in accounting for the general Messianic nature and recognition of the Psalm, as well as less awkwardly assimilating into the presentation of the author Hebrews, since it straight-forwardly takes Psalm 102:25-27 as words of God addressed to the Son (if we accept the meaning of the LXX translation, and then we have all the quotations of 1:5, 1:6, 1:8-9, 1:10-12 and 1:13 as words of God to the Son). This is an approach espoused by Don Carson in his devotional reading notes. Also in an article by Stephen Voorwinde in Vox Reformata. It still has a little awkwardness in that it may seem anachronistic for God to address the Messiah as "God" and "YHWH". However, this is not insurmountable, for similar reasons as in the options above. Also, anachronistic reading is mainly an issue if we have problems with the divine authorship of the Scriptures.
In thinking about this question, I also toyed with another option. Perhaps the author of Hebrews was making a more oblique reference, or that this quotation was not meant to be taken on its own merits, but in combination with the other quotations. Perhaps he did understand that Psalm 102 was referring to God, but that the same things are transferred by inheritance to the Son, since the Son sits at the right hand of God, being exalted to the same authority (cf 1:4 and 1:13). Such an option squares more naturally with the traditional Hebrew reading of Psalm 102 while avoiding the difficulties of not sufficiently distinguishing between God and the Son (which the author clearly does distinguish in chapter 1). However, on balance, given that it is a somewhat convoluted option, it is probably best to go with option 3 above, as a more straight-forward and very acceptable option.