How does one reconcile the changelessness of God with Jesus' incarnation?

This answer to this question is one of the things that sets the The Church of Jesus Christ of Latter-day Saints apart from other Christian denominations. We believe in the Godhead, which is the Father, the Son, and the Holy Ghost as three distinct separate beings.

In the beginning as we know it, God the Father had an exalted physical form, but all his children were spirits, Christ being one of his spirit children. When the plan of salvation was presented, Christ was selected to serve as saviour and redeemer of all God's children, and with god's power, created "the Heavens and the Earth" so that all those who were righteous in keeping their first estate (choosing to follow Christ's plan) in the pre-earth realm could come to earth and receive bodies.

Christ remained a Spirit and was known by the name Jehovah until the meridian of time, when he too was born on earth into a physical body. After his ministry, he died and was resurrected as a glorified exalted being, as the Father is. When we die, we too will be resurrected, every man that lives on the earth, but it is only by righteousness and obedience to God's laws that we too can be exalted and inherit a portion of our father's kingdom.

While different Christian groups might answer this differently, the vast majority today are Chalcedonian Trinitarians. I will answer the question from that perspective.

When it is said that God does not change, this means that God's essence, His nature does not change. The Second Person of the Trinity assumed the Human nature uniting it in His Person. So in this limited sense, you can say that there was a change. However, that change was not a change in the changeless Essence of God.

At the Incarnation, the Divine Nature was united to the Human Nature in the person of Jesus Christ. However, it is a matter of faith that Divine Nature was not altered or changed by this union. From the Formula of Chalcedon:

one and the same Christ, Son, Lord, Only-begotten, recognized in two natures, without confusion, without change, without division, without separation; the distinction of natures being in no way annulled by the union, but rather the characteristics of each nature being preserved and coming together to form one person and subsistence

The divine nature is united with the human nature in person of Jesus Christ, but it is unchanged by this union.

You mention Hebrews 13:8, but this verse needs to be kept in context:

Remember those who rule over you, who have spoken the word of God to you, whose faith follow, considering the outcome of their conduct. Jesus Christ is the same yesterday, today, and forever. Do not be carried about with various and strange doctrines. For it is good that the heart be established by grace, not with foods which have not profited those who have been occupied with them. [Hebrews 13:7-9 NKJV]

Here, Christians are directed to remain in the teachings they have received and not to chase after new one. The Christian teachings remain the same. This is not a passage about the nature of Christ.

This might be a hard one for us temporal creatures to get our head around, and it actually isn't just the incarnation that poses this problem. Viewed from a temporal perspective, we could just as well say: "Before creation, God wasn't a creator - after creation, he became one. How can this be consistent with the idea of God's immutability?"

The only logical solution to such dilemmas consistent with scripture that I am aware of is that God transcends time itself (ie God's eternity is timeless not everlasting time). From God's perspective in the eternal now, He has always been an incarnated God-man and will always be one. Change (not just in this matter, but in any respect) is something that only occurs within the temporal universe above which, God is transcendent:

Concepts of eternity have developed in a way that is, as a matter of fact, closely connected to the development of the concept of God in Western thought, beginning with ancient Greek philosophers; particularly to the idea of God's relation to time, the idea of divine perfection, and the Creator-creature distinction. Eternity as timelessness, and eternity as everlastingness, have been distinguished. Following the work of Boethius and Augustine of Hippo divine timelessness became the dominant view. - Stanford Encyclopedia of Philosophy [emphasis added]