Disappointingly, after a quick Google Search, I found very little on the topic published in the past few days since news of the incredibly discovery were announced. Most news articles found were just recycling the Daily Mail article and were not interested or didn’t know about its possible Biblical connection.(HT reader Gerald Rosenberg.)
Not being an expert myself in ancient Semitic languages, I was very happy to find Jim Davila’s fantastic Paleo-Judaica Blog, where Mr. Davila confirms that it is a stone seal written in ancient Hebrew script.
This find is calling out for more attention. Northwest-Semitic epigraphers, where are you?
Last week's PaleoJudaica post, which links to Daily Mail article that announced the find in English, is here and is quoted extensively by Mr. Benedix.
UPDATE (19 August): Victar Josef Mas, commenter to the Times of Israel article, reminds us that the commander of the seventh-century BCE military fortress excavated at Arad was also named Elyashiv. We have many inscriptions from Arad that mention him (cf., e.g., here and here), including three of his seals, one of which you can see here. The name does not seem to have been uncommon, so I doubt that there is any direct connection between Elyashiv of Arad and the Elyashiv of the Russian woman's inscription.
ANOTHER UPDATE: I have not been able to find the name in any of the other languages. That's not conclusive, but it points to the likelihood that the seal bears a Hebrew name.