Jennifer, I'm a recovering flashaholic, so got rather deep into battery performance a year back. My comments here are not Delorme specific, but general. Mostly targeted at AA and AAA cells, but may furnish some background info.
Loooong story short, I settled on Eneloops (which are NiMH chemistry) in most my AA devices for several reasons: 1) they're safe 2) they don't leak in gear 3) they're easy to recharge 4) they hold charge a good long time 5) they're inexpensive 6) they can dump a lot of energy when they're called upon to do so. But to use these batteries well, one needs a good charger that can every once in awhile refresh the batteries. I use the Maha C-9000, an often recommended charger.
That said, for short backpacking trips (ie, no recharging needed), my Eneloops get laid aside in favor of AA lithium primaries (the NON-rechargeable lithium cells you buy off the shelf in stores) because of their 1) light weight 2) very excellent power.
I explored the rechargeable lithium route briefly (18650s) and on my first attempt, got an overheating cell. That soured me on the experience and I backed out, decided I'd live with lower performance for increased safety. Now there are scads of people who recharge li-ion successfully so I'm definitely an outlier there. My thinking was I'd be using 18650s for headlamps, and given the $$ I've spent on education, I decided I just didn't need that much power <1" from my brain. Again, loads of people use Li-ion on their headlamps with zero problems, so just a personal call.
You don't wear your Delorme on your head, so...probably not an issue for you.
The area you want to be careful with rechargeable li-ions (well, all cells, but be especially cautious with rechargeable li-ions) is a multi-cell device, where you mistakenly insert a depleted or nonfunctional cell, and other full-charge cell(s). That's the sort of situation where the fireworks can happen, when the full cell(s) tries to charge the dead cell. Not good.
Whatever cell chemistry you use, and particularly in multi-cell devices, be certain to learn how to check charge in all your cells and try to match cells for capacity. (A good charger like the Maha for NiMH makes this easy.)