How is Reinforce's name supposed to be said? In season 3 of Magical Girl Lyrical Nanoha, Hayate calls Reinforce "Rein", and near the end of season 2 she renames Dark Book to Reinforce. However, looking at the text, it looks like "re-in-force" rather than "rein-force".

So, I am wondering if "rein-force" is actually how Reinforce is supposed to be called or if it's supposed to be "re-in-force", which however sounds different due to it being said by the Japanese.
Converting my comment into an answer, thanks to textual support from Dusk252's comment...

The character's name in Japanese is リインフォース riinfoosu, which is basically how the English word "reinforce" would be rendered into Japanese. So, if you were talking about Nanoha while speaking in English, you should pronounce the character's name like the English word "reinforce"; that is, "ree-in-FOHRS", not "RAHYN-fohrs" or anything else like that. 

As seen in the topmost speech bubble in this excerpt from chapter 2 of Mahou Senki Lyrical Nanoha Force, Reinforce's name is abbreviated as リイン riin. Thus, you should pronounce the abbreviated form of her name as you would pronounce the correspondingly-abbreviated form of the English word "reinforce": as "ree-in".