Ndtyhij

The planet is called Arda.

So why do they call it "Middle-earth" if there is no planet called Earth to have a middle of?

Because "Arda" is the Quenya word for "Earth."

The phrase "Earth" is actually used repeatedly in the Legendarium, especially in the Ainulindalë, (emphasis mine):

[T]he other Ainur looked upon this habitation set within the vast spaces of the World, which the Elves call Arda, the Earth

[...]

When therefore Earth was yet young and full of flame Melkor coveted it, and he said to the other Valar: 'This shall be my own kingdom; and I name it unto myself!'

But Manwë was the brother of Melkor in the mind of Ilúvatar, and he was the chief instrument of the second theme that Ilúvatar had raised up against the discord of Melkor; and he called unto himself many spirits both greater and less, and they came down into the fields of Arda and aided Manwë, lest Melkor should hinder the fulfilment of their labour for ever, and Earth should wither ere it flowered. And Manwë said unto Melkor: 'This kingdom thou shalt not take for thine own, wrongfully, for many others have laboured here do less than thou.' And there was strife between Melkor and the other Valar; and for that time Melkor withdrew and departed to other regions and did there what he would; but he did not put the desire of the Kingdom of Arda from his heart.

^{The Silmarillion I Ainulindalë}

These are only a few samples; the term "Earth" is used over a hundred times in The Silmarillion alone.

The Quenya name isn't used as much as you might think; it's used less in The Silmarillion than "Earth" is, and then mostly in the first two parts (Ainulindalë and Valaquenta), which are the myths that originated in Aman. From this perspective, it makes sense that the Quenya word be used, since Quenya would have been the original language of translation.

After that, the references to "Arda" are almost exclusively related to events from those earlier stories; again, there's a logic in using the Quenya term.

In other cases, the original myths would have been written in Sindarin, so using a Quenya word doesn't make a whole bucketful of sense. The Lord of the Rings, on the other hand, would have been written in Westron by hobbits; again, not much reason for a Quenya word to be used.

It's worth noting that the continent of Middle-earth has another name in Quenya (not Middle-Arda, sadly). It appears a bunch of times in History of Middle-earth, but only twice in The Silmarillion (emphasis mine):

The Noldor came at last far into the north of Arda; and they saw the first teeth of the ice that floated in the sea, and knew that they were drawing nigh to the Helcaraxë. For between the land of Aman that in the north curved eastward, and the east-shores of Endor (which is Middle-earth) that bore westward, there was a narrow strait, through which the chill waters of the Encircling Sea and the waves of Belegaer flowed together, and there were vast fogs and mists of deathly cold, and the sea-streams were filled with clashing hills of ice and the grinding of ice deep-sunken.

^{The Silmarillion III Quenta Silmarillion Chapter 9: "Of the Flight of the Noldor"}

Tolkien didn't use this phrase very much in his writings, possibly because he could see the future and didn't want to confuse Star Wars fans, or possibly for the same reason he didn't use Arda very much.

The word Middle-earth is just a modernization of the Old-English word Middangeard, which meant "the inhabited lands of Men between the seas". (The same word which Midgard is derived from) It had nothing to due with the planet earth. (Though Tolkien was known to include many language puns in his writings.)

‘Middle-earth’, by the way, is not a name of a never-never land without relation to the world we live in (like the Mercury of Eddison). It is just a use of Middle English middel-erde (or erthe), altered from Old English Middangeard: the name for the inhabited lands of Men ‘between the seas'.

(Letters #165)

"Middle-earth" is a modern version of the word "Midgard" which in Norse mythology is the world that humans live on (there are eight others). Tolkien was known to derive some of the mythology of Middle-earth from Norse mythology (including parts of the Ainulindalë, the creation scene at the beginning of the Silmarillion).

All hail @Jason Baker's excellent answer. I'd like to add one additional very important point:

Tolkien reported that the original inspiration for everything in his Legendarium was in 1913 when he was studying Anglo-Saxon literature and came across lines in the Crist written by Cynewulf:

éala éarendel engla beorhtast / ofer middangeard monnum sended

which can be translated:

"Hail Earendel, brightest of angels, over Middle-earth to men sent"

He wondered who or what Earendel could be and spent the next sixty years inventing legends, stories, languages, a whole world to explain it. It's Middle-earth because that's what it was in its deepest roots.