I think most of the issues I have with resting in 5th (and D&D in general really) is that there is too much of a gap between what is defined as a short rest and what is defined as a long rest. (I also feel like 5E is pretty relaxed on all of the things you can do during a long rest and still be rested.) I like the idea of short rests being shorter, but I'm still fine with long rests being the length they already are.
I've been playing around with the idea of introducing "medium rests," but it's currently just a rough idea.
The format might look strange to this; I just copied from a conversation I was having elsewhere:
"I think 5E's rests are too long, but I think it's still based on some different assumptions than 4E. I know it's not in the book, but I could see 10 minutes for a short and then somewhere between 1-4 hours for a long rest. In either case, if you change long rests to only being 1 hour, I'd consider using something similar to the half-spell-slot thing mentioned in the DMG. If it's not too complicated for people to understand, I'd propose having short rests, "medium" rests, and long rests.
Short Rest - 5 (or 10 minutes), refreshes things that refresh during a short rest.
Medium Rest - 1 hour or more, benefits of a short rest plus spellcasters can recover a number of spell slots equal to half of what they normally get (round down). For example, a third level sorcerer has 4 first level slots and 2 second level slots, so -during a medium rest- that same sorcerer could recover 2 first level slots and 1 second level slot. Spell slots above 5th level cannot be recovered during a medium rest, but may be recovered using other means such as a sorcerer's spell points. Eating a meal and/or using a healer's kit to bandage wounds might allow recover of a hit die or regaining HP as though a hit die were spent.
Long Rest - 8 hours, all resources are refreshed as normal for a long rest. See page 186 of the PHB for details."
What I had in mind for variable times was based on similar ideas to what Flamestrike was saying. A short rest might only be 5 minutes in a relatively safe place. The 10 minute time could be deemed necessary by the DM in areas where an element of danger requires the party to take precautions for safety and security. The same concept is applied to medium rests. I had considered allowing the party to opt to still use the shorter times in dangerous areas in exchange for a higher chance of random encounters or a higher chance of being interrupted by attackers.
Medium Rests are a slightly modified version of how the DMG defines long rests if the heroic options are used.