The NET translation of Psalm 110:5 takes the approach of capitalizing the word to clarify to whom אֲדֹנָי is referring:
O sovereign LORD,20 at your right hand
he strikes down kings in the day he unleashes his anger. (Psalm 110:5 NET)
Capitalizing LORD treats the word as the tetragrammaton, as is found in many medieval Hebrew manuscripts, in order to show it is referring to God.
The translator notes in the NET Bible explain there are 3 possible interpretations:
(1) As pointed in the Hebrew text, this title refers to God (many medieval Hebrew mss read יְהוָה, yehveh, “Lord” here). The present translation assumes that the psalmist here addresses the Lord as he celebrates what the king is able to accomplish while positioned at God’s “right hand.” According to this view the king is the subject of the third person verb forms in vv. 5b-7.
(2) Another option is to understand the king as the addressee (as in vv. 2-3). In this case “the Lord” is the subject of the third person verbs throughout vv. 5-7 and is depicted as a warrior in a very anthropomorphic manner. In this case the Lord is pictured as being at the psalmist’s right hand (just the opposite of v. 1). See Pss 16:8; 121:5.
(3) A third option is to revocalize אֲדֹנָי (’adonay, “Lord”) as אֲדֹנִי (’adoniy, “my lord”; see v. 1). In this case one may translate, “My lord, at his [God’s] right hand, strikes down.” In this case the king is the subject of the third person verbs in vv. 5b-7.
Option 3 requires rewriting the text by replacing אֲדֹנָי with אָדוֹן. One difficulty in translating the verse into English (or Greek as in the LXX) is the Psalmist used both אָדוֹן and אֲדֹנָי:
The LORD says to my Lord (אָדוֹן):
“Sit at my right hand,
until I make your enemies your footstool.”
לְדָוִד מִזְמֹור נְאֻם יְהוָה לַֽאדֹנִי שֵׁב לִֽימִינִי עַד־אָשִׁית אֹיְבֶיךָ הֲדֹם לְרַגְלֶֽיךָ׃
The Lord (אֲדֹנָי) is at your right hand;
he will shatter kings on the day of his wrath.
אֲדֹנָי עַל־יְמִֽינְךָ מָחַץ בְּיֹום־אַפֹּו מְלָכִֽים׃
אֲדֹנָי in verse 5 is the emphatic form of אָדוֹן verse 1, which the Gesenius' Hebrew-Chaldee Lexicon understands is used only of God. Charles Ellicott comments on the difference:
We are naturally tempted to understand this as still the king whom the first verse placed at Jehovah's right hand. But the word for Lord here is Adonai, which is nowhere else used except of God. Moreover, God throughout has as yet appeared as the active agent. It is He who stretched out the sceptre and conferred the office of priest; and hitherto the king has been the person addressed. It is therefore necessary still to consider him as addressed, and suppose that the change of position of Jehovah from the king’s right hand to his left is simply due to the usage of the language. To sit at the right hand was an emblem of honour, to stand at the right hand was a figure of protecting might (Psalm 16:8; Psalm 109:31); and the imagery of a battle into which the song now plunges caused the change of expression.
While the NET is in the minority in rendering the word as LORD, clearly the Masoretic Text preserves a difference from "Lord" in verse 1 and "Lord" in verse 5 which should be conveyed in translation. This, coupled with the fact some medieval manuscripts have the tetragrammaton in verse 5 is strong evidence in support of this approach. Finally, since both the Masoretic Text and the medieval manuscripts were produced during the Christian Era, any bias in translation would likely be against seeing the Psalm as a reference to Jesus.