何とやら is a placeholder (like "you-know-what(/who)", or "what's-his(/her/its)-name"), used in place of names you can't remember, or words and phrases you don't want to mention explicitly.
Notably it has a distinct use as a replacement for a part (often the latter part) of established expressions, like proverbs and idioms. （Examples: 「ちりも積もれば何とやら (for 山となる)」, 「枯れ木も山の何とやら (for 賑わい)」, 「触らぬ神に何とやら (for 祟りなし)」. I think the reason for its employment, most of the time, is the psychology "The less said of a cliche, the better," rather than economy, forgetfulness, or polite euphemism.)
In this case the idiom (of sorts) they had in mind was probably "遠足前の小学生", a variation of the more accurate "遠足前日の小学生" - "a grade school kid on the eve of a field trip" - used for when you are too excited about the next day's event to go to sleep. (Or less likely "遠足前(日)症候群", which refers to the same thing.)
My translation effort:
"I can rarely sleep the day before a competition - the proverbial kid on the eve of a field trip. I'm weak-nerved."