I've caved for almost two decades, and for more than half that time I used *only* a construction helmet (with a 3-point chin strap I added).
My original caving club in New Zealand has a bunch of construction helmets with chin straps and Premier cap lamps that they've been loaning out to new cavers for years - actually decades. There's never been a problem with this that I'm aware of. They have good chin straps, and are frequently used for vertical caving by adults and younger cavers alike. Of course, in New Zealand there isn't a lawyer lurking under every rock waiting to sue you...
I know, I know... construction helmets - aka hard hats - are designed to resist a blow from above, but not a side intrusion. But let's face it, a caving helmet's main job is to stop you bumping your head, becoming distracted, and then doing something that results in a real injury - especially when on rope. A construction helmet and a climbing helmet probably offer similar protection if falling rocks hit you from above. A blow from the side is most likely to be you hitting the rock, not a big rock hitting you. A properly-fitting construction helmet with chin strap should be adequate protection against that.
A good climbing/caving helmet does have advantages over a construction helmet though. They tend to fit more securely and comfortably, and have a lower profile on your head. The only disadvantage is cost. In the US it's not such a big deal, due to availability and the size of the consumer market. You can get a good helmet anywhere from about $40 to $80. In NZ, you can't touch one for much under $100. Between import duties and GST (sales tax), prices are horrific. I saw a Petzl Duo (non-LED) in a store there last year for $180!
Since I started using a *proper* helmet, I've never looked back. However, I've also helped on plenty of novice trips in the US where kids and adults alike have worn bicycle helmets or construction helmets. A couple of years ago, I purchased a set of 3 HB climbing helmets and 3 Petzl Zoom headlamps for my grotto to use as loaners. They cost the grotto about $180 all up, and there is no charge for their use (or any requirement for donations). Another liability thing, I believe. They recently added a child's size Petzl helmet to their loan department too.
I agree with previous posts that talk about making sure novice cavers have a positive first-time experience. If possible, and as applicable to the particular cave, this should include providing a comfortable helmet with chin strap (climbing or otherwise), and a good headlamp. It also means checking everyone has the appropriate clothing and footwear. 8)
>>>By the way, Martha made a comment about not entering mines. I guess in her region, like many regions in the US (so I'm told), mines are not too stable. Too much dirt, not enough solid rock - something like that, perhaps? I've only been in a few mines in the US, so I'm not really sure. The reason I'm mentioning all this is beacuse I spent years exploring abandoned mines - mainly relics from New Zealand's gold rush in the late 1800s. In fact, mines got me into caving.
In the past I've "discovered" a few unstable mines, and promptly exited them again. But I've been in a bunch more that have been fine, as long as one is careful and knows what to look for (a little like caving, really). I've been on rope in numerous mines, photographed them, mapped a couple, used some for rescue training, and even taken Scout groups through one or two. All I'm saying is that the "mine paranoia" exuded by many cavers in the US and other countries is perhaps not always justified. In fact, Joe Public in the US probably thinks "Stay Out, Stay Alive" also refers to caves...