Consulting Daijisen, っこ
means …のこと
or …すること
(as in 〜慣れっこ
"being used to" etc) according to sense (1)(1).
Daijisen also lists in sense (1)(2) "an action done mutually with two or more people" as in とりかえっこ
"an exchange" or sense (1)(3) "an action where two or more people compete" as in 駆けっこ
"race"/"sprint".
A common usage of っこなしにしよう
might be like this example from the Kenkyuusha dictionary:
このことは誰にも言いっこなしにしましょう. お分かりいただけますね.
Let us keep it (strictly) to ourselves. I hope you understand.
〜っこなし
= 〜(のことなし・することなし)
which I think here means "not (do)ing".
〜にする
= "go with"/"decide to" and 〜にしよう
means "let's go with"/"let's decide to".
〜逃げっこなしにしよう
= 逃げ(のことなし・することなし)にしよう
= "let's go with not running away"/"let's decide on not running away".
So from there:
「いいか、この最後のローソクが消えたら、本当のお化けが出るかもしれん。だが、どんなお化けが出ようと、お互いに逃げっこなしにしよう」
"Look here, if this last candle goes out, there could be an actual Obake that appears. But, whatever the sort of Obake that should appear, let's together decide on not running away."
「いいとも。どんなお化けが出るか、この目で、しっかり見てやろう」
"All right. Whatever the sort of Obake that appears, let's sit tight and witness it with our own eyes."
(この目で見る
= "[see/witness] [firsthand/with our own eyes]" etc.)
Consulting the 日本語文型辞典
, another common usage of っこ
in the form of っこない
is to strongly negate the chances of something happening. It's used in informal speech and is close to 絶対…しない
, …するはずがない
and …するわけがない
, for example:
いくら彼に聞いても、本当のことなんか言いっこないよ。
"However much you ask him, [there's no way he'll/he absolutely won't] tell the truth."
Edit: Tried to fix various mistakes.