Abram

Ab means chief, principal, and comes from a primitive root word meaning father, chief (The same word that Yeshua used when he cried Abba, Father.)

Ruwm means exalted, haughty, and comes from a primitive root word that means to be high above, to rise or raise, to bring up, to exalt or be exalted.

Abraham

Ab is the same; nothing changed.
Raham means a multitude.
So, Abraham went from being a father exalted (alone), to being the father of a multitude.

As for me, behold, my covenant is with you, And you will be the father of a multitude of nations. No longer shall your name be called Abram, but your name shall be Abraham; for I have made you the father of a multitude of nations. I will make you exceedingly fruitful, and I will make nations of you, and kings will come forth from you. I will establish my covenant between me and you and your descendants after you throughout their generations for an everlasting covenant, to be Elohim to you and to your descendants after you. I will give to you and to your descendants after you, the land of your sojournings, all the land of Kna'an, for an everlasting possession; and I will be their Elohim. (Bereshit 17:4-8) 

This name change did not change the basis of who they were; instead, it was built upon.

Sarai

Sar means chief, captain, general, lord, keeper. It comes from the word sarar which means to be or act as a prince, to rule, a head person (of any rank or class) a captain, chief, general, governor, keeper, lord, ruler, steward. It comes from primitive root word that means to have or exercise dominion, to rule.

Ai is a feminine suffix, which makes any word that it is attached to become the feminine version of that word. It is also associated to barrenness. (Bereshit 11:30)

Sarah

Sar is the same~nothing changed.

Ah is also a feminine suffix, and changes the meaning lady, princess, queen, and takes away the barrenness.
So, Sarah went from a barren leader to a leader of royalty. (Bershit. 17:16)

Then Elohim said to Abraham, As for Sarai your wife, you shall not call her name Sarai, but Sarah shall be her name. I will bless her, and indeed I will give you a son by her. Then I will bless her, and she shall be a mother of nations; kings of peoples will come from her. (Bereshit 17:15-16)

Both before the name change and after, Sarah's status in relation to Abraham did not change. Sarah did not lose her 'royalty' status, rather she was promoted in it; she would have an heir to carry it on.