Laverman is a Dutch surname. According to www.meertens.knaw.nl it is either a professional surname or a toponymic one. It doesn't give more info than that. The toponymic one i can't understand, but it probably would be a 'man from Laver'. But where is Laver? Is it a corruption of the French Le Havre? I don't know. I'll try to find that out later.
The occupational surname Laverman leaves me a better chance to understand. I used the dictionary to look up the verb laveren , which has three meanings. One of them is 'to tack' (a sail term) and the other two are 'to stagger, to reel, to totter' and 'to steer a middle way, to tack, to navigate'. The latter two in my opinion are just derivations from the first one (they are just in a way metaphores for a person's actions and behaviour). Thus, Laverman as an occupational surname would probably mean a person responsable of this task, i.e. tacking (the one at the steer, thus like 'steerman').