This really gets down to the purpose and audience of your hero images. Thankfully, Midas' question in the comments and your answer provides some insight and I feel that maybe you need to change the approach a little.
I'm assuming that your site is for a travel agency, tourist destination, or something similar. If it was me, I wouldn't use a single hero image that gets changed once a week, or month or every few months.
Instead I would use a variety of images and create a slideshow that just loops through them.
This way you could change the slideshow as required (to suit a change of season, a holiday period, etc). In other words, it's not changed at a set interval like weekly, monthly, etc. Instead you change the slideshow to suit the industry you're in. If Christmas is a busy period, then change the slideshow to suit. If Summer is a busy period, change it again. And for variety you can change it regularly for no particular reason too.
This ensures there is enough variety in the images to appeal to a broad spectrum of people. All you need to do is collect a library of images and decide how many you want per slideshow and how fast they loop through the images.
Hope this helps. Good luck.
Okay, now that you provided a link, I'd like to add the following.
For starters I like the look of the site. Well done.
Because of the panoramic nature of your hero image and the way you have overlaid text, I agree that a slideshow or animated gif or anything that distracts the user from your text is not a good option.
However, my earlier comments regarding timeframe still stand. I wouldn't be restricted by a schedule as such. I would have a library of images ready and then rotate them through when appropriate based on what's going on at the time. This doesn't just mean a change of seasons or the occurrence of holidays, it may be an event. For example, the Soccer World Cup and the Olympic Games are events that attract tourists for a totally different reason.
So, in a nutshell, I would get the marketing people and other stakeholders together to decide what particular events (be it seasons, national holidays, sporting, etc) are typically drivers for people wanting to take holidays. That offers some reasoning behind how often you change the image, rather than some arbitrary schedule.
Otherwise it really just gets down to personal preference - albeit biased by whether you're a marketing person, a business owner, a UX advisor or whatever.