If it were my son I would get some McNett seam grip and use it to stick two round patches, one on each side of the hole. I’ve repaired some gear with seam grip and it lasts a long time. I’ve not had quite the same longevity with Tenacious tape. It would be for piece of mind more than anything else since she is pretty light and an ember would have singed the fibres. A tear would be more of a problem. YMMV.
Don't let life get in the way of living.
If it was my daughter, I wouldn't let her use that hammock anymore. I won't even let kids use any hammocks made from fabric less than 40D / 1.5oz. Yes, the hammock might hold up for a while, and even if it rips your daughter might be fine. But if something bad happens to her because you let her use a damaged hammock, you'll probably curse yourself for the rest of your life. A new hammock doesn't cost much. Certainly a lot less than a doctor. Make some new gear from the hammock, like a gear hammock, stuff sacks, a Wooki-style underquilt, ... There are lots of possibilities.
While it's true that any alteration of the fabric of the hammock will reduce its structural integrity, you have to consider how big the hole is, what type of fabric, and where it is located. One of the reasons we use rip-stop is to prevent any hole from spreading. It looks like that has been accomplished from your photo.
If concentrated force is going to be applied to that location I'd recommend retiring it but if it's a general force, you'll probably be fine with a patch like Tenacious tape (or sewing a patch) to help distribute the weight even further.
I accidently poked a hole in a brand new DIY bridge hammock right near my shoulder at the edge. After the cursing stopped, I checked to see what the hole was going to do when I was in it. After applying weight (carefully) and probing the hole, it looked like it wasn't going to do anything so I patched it with some Tenacious tape and it's been fine ever since. Obviously, your mileage may vary.
We always assume some risk when we camp, be it from falling from a hammock, a widow maker, animal attack or fire. We want to be smart about it and evaluate the actual risk. The standard rule of "don't hang higher than you are willing to fall" seems applicable in this case. I've been dumped on my butt as probably has everyone else at one time or another. If it makes you more comfortable then turn it into a gear hammock or stuff sack.
Interesting thread. I'll share my experience. My favorite hammock material is no longer available 1.4 PolyD. I love the feel of it as it has the perfect amount of stretch for me. That said, it was a pretty fragile fabric. I made a couple hammocks out of it and each seemed to develop holes or wear areas after a year or so of use. The first one I patched with some tenacious tape on each side. The tape has held up well and I've slept in that hammock without issue probably 5 times since it developed the hole. From you photos, I'd say my hole was bigger than yours and I weigh 180 lbs. for reference. From what I can see, the tape has prevented any further damage.
All that said, once my hammocks get a hole they no longer go on backpacking trips with me. I will use them to sleep in the backyard over a nice soft lawn, or for occasional lounging, but if I have to trust it for a trip, I personally don't want it to have any holes in it. Based on my experience, I'd probably be fine for awhile using it with a hole. I just always have it in the back of my mind that it is now more susceptible to failure and I want to avoid that in the mountains.
I can't really tell you what to do with your hammock, but hopefully my experience gives a little more information for you to make a decision with.
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