The Elder Futhark was in use ca 100-800 CE, mostly in Scandinavia, but also in (modern) Germany, Eastern Europe and Britain. During this period, the Proto-Norse language changed tremendously. Transliterated Old Norse, which came after, is somewhat readable today by mainland Scandinavians. Proto-Norse from 100 CE is almost totally unintelligible. Thus, asking for the word for "carve" written in the Elder Futhark is not a very well-defined question, as the Futhark is simply another type of alphabet - you also need to specify a language.
But suppose we would be satisfied with any type of Proto-Norse/Old Norse? The Rök Runestone was created at the end the period, and includes a part of how it was created. Furthermore, the Sparlösa Runestone is just a bit younger. The Rök stone mainly uses the Elder Futhark, the Sparlösa stone mainly the younger, but they both mix a bit. Interestingly, they also both uses a different word than would later be used for the carvings done using the younger Futhark: instead of "risti", they used "faþi".
Thus, for late Proto-Norse rendered in the Elder Futhark, the word you'd use when writing runes would be
However, on the Rök stone, it was actually written
For comparison, here is a rendering of "risti" in the Younger Futhark, which is what Halvdan would probably have written in the Hagia Sofia:
(This is one of the most common word on all rune stones, as they were raised by the thousands in the area around Stockholm in the 11th century CE, often using a highly standardised message).