The mercy seat is an incredibly dense and intertextual image, with a long history in Christian culture. Below is an illustration by Michiel van der Borch from a 1332 copy of the "Rhyme Bible" (which despite the name, does not rhyme and is not a Bible), showing many facets of the concept. 1
Here, God the Father sits on a throne that might look a bit like the Ark of the Covenant; he is holding the crucified Christ in his lap, and the Holy Spirit is also present as a dove. This is actually a fairly common artistic motif for showing the Trinity, often called the "Gnadenstuhl" image (German for "grace stool" or "mercy stool"). Anyone who thinks "oh, now that makes total sense" can stop reading now. For everyone else...
Biblically, the original element is the Ark of the Covenant. From descriptions in the Torah2, we learn that the Ark has a golden lid called the kapporeth (כַּפֹּ֫רֶת). The lid therefore covers the tablets of the Law, kept inside the Ark. It also has two angelic figures mounted upon it, who stretch out their wings above the Ark: this space between the wings was particularly associated with the presence of God, and seen as a sort of earthly shadow of God's heavenly throne (see e.g. 1 Samuel 4:4).
Of course, the whole Ark was meant to be kept in a place of sanctity, the summit of the system of ritual purity. Ultimately, only the High Priest could enter the Holy of Holies in order to offer a sacrifice on the Day of Atonement, Yom Kippur - he would sprinkle the blood of a bull on the kapporeth, as the climax of all ceremonies and observances directed towards the seeking of atonement. The similarity between the words "Kippur" and "kapporeth" is no accident: the concept of "covering" is etymologically and philosophically linked to the idea of having sins "made better". In the Greek Septuagint, the word used is hilasterion (ἱλαστήριον), which more directly means something like "propitiation", "atonement", or "expiation", and appears again in the New Testament. (See: "How should ἱλαστήριον (hilasterion) be translated in Romans 3:25?" on BH.SE.)
A very direct connection is made by the author of Hebrews in chapter 9. Verse 5 repeats the idea of the hilasterion. But now Christ is come to be both High Priest and sacrifice.
But when Christ came as a high priest of the good things that have come, then through the greater and perfect tent (not made with hands, that is, not of this creation), he entered once for all into the Holy Place, not with the blood of goats and calves, but with his own blood, thus obtaining eternal redemption.
Nor was it to offer himself again and again, as the high priest enters the Holy Place year after year with blood that is not his own; for then he would have had to suffer again and again since the foundation of the world. But as it is, he has appeared once for all at the end of the age to remove sin by the sacrifice of himself. (verses 11-12 and 25-26, NRSV)
The mercy seat, which was the sign of God's presence and the place of atonement, is now identified with the ultimate act of atonement performed by Christ, in his own person. If we look back at the illustration above, we can now understand Christ being received as a sacrifice on the throne/altar/Ark. He has replaced the old system of purity under the Law, and is being crucified symbolically "above" the Law.
Moreover, God's presence in the world is now the body of Christ himself - "God with us", the Word made flesh - and for that reason the mercy seat is also sometimes associated with the Eucharist. As Peter Turner points out in a comment above, the Ark may stand for Mary, which is another link to the reality of God's presence due to his Incarnation.
Because the Ark is a sort of earthly throne for God, mirroring the heavenly throne, there is some cross-pollination with other Biblical throne imagery. In Hebrews 4:14-16 is the famous language of the "throne of grace", associated with Christ as High Priest. There is also the throne of Revelation 4. The song lyrics you refer to in the question are clearly drawing partly from this chapter (throne, rainbow, lightning, "Holy, Holy, Holy") as well as the following one. The mercy seat reference is:
Worthy is the Lamb who was slain
Holy, Holy, is He;
Sing a new song to him who sits on
Heaven's mercy seat.
This comes from Revelation 5, where Christ is presented as the slain Lamb:
They sing a new song: "You are worthy to take the scroll and to open its seals, for you were slaughtered and by your blood you ransomed for God saints from every tribe and language and people and nation; you have made them to be a kingdom and priests serving our God, and they will reign on earth." [...] "To the one seated on the throne and to the Lamb
be blessing and honor and glory and might forever and ever!" (verses 9-10 and 13, NRSV)
So the one who is sitting on the mercy seat, or throne of Heaven, is God (the Father). The slain Lamb, who is God (the Son), is the offering in his own person - this may also be seen as on the mercy seat, and as well, he is "at the right hand of the Father". But in a way, Jesus also is the mercy seat, as in this sixteenth-century exhortation:
Brethren, we haue Iesus Christ the righteous and aduocate with the Father, he is the mercie seate of our sinnes.3
This makes sense if we read the image, in a more general way, as something like "the locus of God's sacrificing presence".
1. Museum Meermanno (Den Haag), MS 10B21, folio 118r. Scanned copy here. The Rijmbijbel was written in 1271 by the Flemish poet Jacob von Maerlant, based on the twelfth-century Historia scholastica Biblical paraphrase of Petrus Comestor.
2. Exodus 25:18-22, 26:34, 30:6, 31:7, 35:12, 37:6-9, 39:35, 40:20; Leviticus 16:2-15; Numbers 7:89.
3. Oxford English Dictionary, 3rd edition, under "Mercy-seat" sense 2a. John Marbeck, A book of notes and common places with their expositions, 1581, p18.