State changing its name within the United States

Is this first time an initiative was taken for a state to change its name after becoming a state within the Union.?

No, 1989 was not the first time. In 1947 a similar resolution was proposed and defeated in North Dakota. See below Origin of State Names Also see Somewhere in America:

At least four times since the end of the Second World War - 1947, 1983, 1989 and 2001 - the name change debate has made headlines.

Although Somewhere in America (which also contains some background on the ND name-change movements) mentions four times, apparently none of the initiatives ever got as far as becoming a resolution that was debated by the ND Legislature besides those of 1947 and 1989.

The 1947 initiative is also mentioned here:

An attempt to drop the word North from the state name was defeated by the 1947 Legislative Assembly. Again in 1989 the Legislature rejected two resolutions intended to rename the state Dakota.

Perhaps records from the ND legislature are available online on one of the state's sites, which would give us an authoritative source for the 1947 initiative, but I have not yet done an extensive search for that.

Did a state ever actually change its name after admission?

It appears that no admitted state has ever changed its name "for the sake of it", when no change of territory was also involved. After considerable research, I found find this site: Origin of State Names, which appears to have some credibility, and mentions the North Dakota initiatives, but there is no mention of any name change of any of the other states.

There have been attempts made to change the state name by dropping the "North" and renaming the state simply "Dakota," but these resolutions were defeated in 1947 and again in 1989.

Although in T.E.D.'s excellent answer he cites the examples of Maine and West Virginia, these were not mere name changes, but changes in the borders of pre-existing states that resulted in the creation of new states, which of course required new names.

Is there a federal process for changing a state's name?

At first blush, such a matter would not be a federal process, nor fall under the jurisdiction of the Federal Government at all: In the body of the US Constitution, there is no reference to the question of changing a state's name, assuming it involves no change to the state's territory:

Article IV, Section 3:

New States may be admitted by the Congress into this Union; but no new State shall be formed or erected within the Jurisdiction of any other State; nor any State be formed by the Junction of two or more States, or Parts of States, without the Consent of the Legislatures of the States concerned as well as of the Congress.

The Congress shall have Power to dispose of and make all needful Rules and Regulations respecting the Territory or other Property belonging to the United States; and nothing in this Constitution shall be so construed as to Prejudice any Claims of the United States, or of any particular State.

Changing the name of an already admitted state (specifically the topic of this question), which has no bearing at all on the territory of that state, any other state, or the territory of United States at large, is irrelvant to Article IV, Section 3, which specifically discusses matters impacting territory. So if it's possible at all, it assumedly would be in the province of the States themselves, as per Amendment 10:

Amendment X:

The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.

The process would normally involve a referendum and an act by the State Legislature and/or amendment to the State's Constitution.

Although the reference to North Dakota indeed cites the Enabling Act of 1889:

In order to legalize the proposal, officials would have to alter the state constitution and the federal Enabling Act

If correct, that was because the Enabling Act of 1889 specifically designates the name North Dakota when the state was formed and admitted:

Enabling Act of 1889:

Be it enacted by the Senate and House of Representatives of the Un ited States of America in Congress assembled That the inhabitants all that part of the area of the United States now constituting the of Dakota Montana and Washington as at present described may become the States of North Dakota, South Dakota Montana, and Washington respectively as hereinafter provided.

Here are some good references regarding a movement to change the name of Rhode Island that appear to confirm this - it is a decision to be rendered by the state itself:

Rhode Island Weighs Using Shorter Official Name

After years of defending the state’s name, the State Senate and House of Representatives voted overwhelmingly last week to allow a referendum asking voters whether to shorten it by seven syllables, to State of Rhode Island. On Tuesday the Senate could adopt the House’s bill, paving the way for the referendum.

Rhode Island Closer to Changing State Name Over Slavery

The country’s smallest state has the longest official name: “State of Rhode Island and Providence Plantations.”

A push to drop “Providence Plantations” from that name advanced farther than ever on Thursday when House lawmakers voted 70-3 to let residents decide whether their home should simply be called the “State of Rhode Island.” It’s an encouraging sign for those who believe the formal name conjures up images of slavery, while opponents argue it’s an unnecessary rewriting of history that ignores Rhode Island’s tradition of religious liberty and tolerance.

The bill permitting a statewide referendum on the issue next year now heads to the state Senate.

It appears from these references that this is a matter for the state itself, not the Federal Government, which has no mentioned involvement in the process, and which is in no way impacted by such a change of name, since it does not involve a change of territory.

Regardless, it appears that Rhode Island never did change its official name, as indicated by language of the preamble to the current constitution of the state, posted on RI's official state website:

WE, the people of the State of Rhode Island and Providence Plantations...

I found nothing regarding any admitted state actually changing its name, when no territorial changes were also involved.

Could we claim that changing the name of the state without changing its territory, population or mode of government would require readmission to the USA because it has effectively become a new state? No: If such was the case, changing the name of a state would be patently illegal, as it would constitute secession.

There is however, one legal argument that can be made that would require an act of Congress to authorize the changing of any state's name, as follows: Like North Dakota, each state was admitted to the Union with a specific name. Here are two recent examples:

An Act to provide for the admission of the State of Alaska Into the Union:

The State of Alaska is hereby declared to be a State of the United States of America, is declared admitted into the Union on an equal footing with the other States in all respects whatever...

An Act to Provide for the Admission of the State of Hawaii into the Union:

The State of Hawaii is hereby declared to be a State of the United States of America, is declared admitted into the Union on an equal footing with the other States in all respects whatever...

If so, perhaps in addition to action on the state level, as mentioned above, an act of Congress would be required to change the name of a state that was previously admitted under another name, similar to what's mentioned regarding North Dakota. It would be akin to the process involved when an individual changes their birth name to a new name.