The inspiration was presumably drawn from Leviticus 24:16:
וְנֹקֵ֤ב שֵׁם־יְהוָה֙ מ֣וֹת יוּמָ֔ת / רָג֥וֹם יִרְגְּמוּ־ב֖וֹ כָּל־הָעֵדָ֑ה
wĕnōqēb šēm-yhwh môt yûmāt / rāgôm yirgĕmû-bô kol-hāʿēdâ
Whoever blasphemes the name of the LORD shall surely be put to death. / All the congregation shall stone him. (ESV)
This comes into the Greek (LXX Rahlfs | NETS):
ὀνομάζων δὲ τὸ ὄνομα κυρίου θανάτῳ θανατούσθω· λίθοις λιθοβολείτω αὐτὸν πᾶσα συναγωγὴ Ισραηλ·
Whoever names the name of the Lord — by death let him be put to death; let the whole congregation of Israel stone him with stones.1
The Hebrew נקב (nqb, "blasphemes") is being used in a sense that is not typical for this word, which normally means "to pierce" or, in a neutral sense, "to designate". Modern lexicons generally agree that the context requires that it here indicates:
a disrespectful or inappropriate treatment of Yahweh’s name.2
On the other hand, at least some commentators maintain that the word simply means "pronounce", at it does in other contexts (e.g. Isa 62:2).3 This may explain the LXX choice of "to name" but it hardly changes the meaning in context, as Milgrom understands this as a circumlocution to avoid the collocation of the divine name and the verb "to curse".
Whatever the original intention of Leviticus, the prevailing understanding in the First Century (likely influenced by the LXX translation) apparently took Jesus's claims to fall under the rubric of Leviticus 24:16 (see also John 8:58-59). This is perhaps most clear in the explanation offered when Jesus questions their motivation for stoning (10:33b):
ὅτι σὺ ἄνθρωπος ὢν ποιεῖς σεαυτὸν θεόν.
[It is] because you, being a man, make yourself God.
1. "By death...put to death" and "stone him with stones" represent normal Hebrew that has been awkwardly reproduced in the Greek and then translated as-is by the NETS. That piece of it should not be understood as a deviation from the Hebrew although it may seem to be when comparing NETS to a smoother translation such as the ESV.
2. Gary Alan Long, "נָקַב (nāqab)", in New International Dictionary of Old Testament Theology and Exegesis, ed. by W.A. VanGemeren (Zondervan, 1997). BDB has it as a separate root "to curse" (perhaps a by-form of qbb).
3. Jacob Milgrom, Leviticus 23–27 (The Anchor Yale Bible; New Haven: Yale University Press, 1974), 2114-2118. He points out that verse 11 (wayyiqqōḇ... ʾeṯ-haššēm wayĕqallēl | "He pronounced [nqb]....the name and he cursed [qll]") associates the pronunciation with the curse; verse 16 now depends on that association in order avoid using qll together with yhwh.