The first part in meridies meant 'share' (of the day) and it's cognate with the verb mereor `earn 'acquire', merenda, Gr.meiromai (receve one's portion) and so on. https://en.wiktionary.org/wiki/μείρομαι#Ancient_Greek http://starling.rinet.ru/cgi-bin/response.cgi?single=1&basename=%2fdata%2fie%2fpokorny&text_number=1794&root=config
Disimulation from a previous *medidies might be pointless. I do not remember such a thing in the classical Latin. Usually 'd' is replaced by an 'l': lacrima from Old dacrima, lingua from Old Latin *dingua.