First and foremost, the correct pronunciation of יישר כוח is Yishar Koach, and similarly Yishar Kochacha. I have read that in Chabad the common pronunciation is yeyasher, but among Hebrew speakers, Yishar is standard and Yeyasher would be interpreted as a blunt error. The ambiguity is probably due to the fact that without Niqqud, יִישַׁר Yishar is indistinguishable from the much more common יְיַשֵׁר Yeyasher, which means "will make straight".
I'm adding some information based on the Hebrew Academy's figures of speech page - it is naturally in Hebrew, but online translation can confirm if necessary.
The origin of this figure of speech is indeed Shabbos 87a as noted in Scimonster's answer.
The common interpretation of the word yishar is of the Hebrew root ישר which is related to straightness and honesty (as in Num 23:27). However, linguist Ze'ev Ben-Haim's opinion is that it is from the Aramaic root שרר which is related to strength and persistence (the modern Hebrew word for muscle, שריר Shrir, is of the same root). This opinion was accepted by the Hebrew academy's historical dictionary, which included the interpretation of יִשַּׁר in this context as "let it be strong", and thus the blessing can be translated as "let your strength persist".
With this interpretation, יישר כוחך "let your strength persist" actually has almost the same meaning as חזק וברוך "strong and blessed". The similar יישר כוח, commonly abbreviated Shkoyach, means "let (someone's) strength persist" and it is understood as a generic blessing.
Finally, the similar blessing חזק ואמץ (Chazak Ve Ematz) means "Be strong and brave", and it comes from the Torah: It is in Moses's blessing to Joshua in Devarim 31:7, and in many other places, in a similar meaning.
Readers of Hebrew may find more information in the Hebrew Wikipedia article.