I say "friendlier" because I don't know if there is a complete eco-friendly way of wrapping. Maybe just planting trees? That would help for sure. I did want to share some ideas though - please feel free to add your own in the comments!
Option #1 : Bag it but skip paper gift bags. Paper gift bags tend to be expensive and normally require some tissue paper as well. Normally they are not used more than once because they show so much wear. Instead, purchase re-usable shopping fabric or canvas bags and stick your presents right in them. The gift receiver gets not only the gift but a pretty reusable bag to use for their shopping. PLUS - re-useable bags are addicting. Get someone hooked. The one shown above is from Baggu - I am an avid collector. Other fun brands are Maptotes & Envirosax. And PS - the tree stamp on the tag can be found here.
Option #2 : Use the right size box for your gift. You know when you have a t-shirt to wrap but only a big sweater box on hand? To make it look "full" it's easy to stuff the whole thing with tissue paper. DON'T. Plan it out and stock up on boxes to fit your gifts. If the item fits snug enough, there will not be any room, or any need for tissue paper.
Option #3 : Package your presents in patterned storage boxes. No need to wrap it up in paper, just tie a bow and call it done. Look for patterns that say "festive" without screaming "holiday". Festive boxes can be used to store stuff on a shelf all year long. It's another two gifts in one. The one above currently holds my greeting cards and I bought it at Ikea.
Option #4 : Buy recycled paper to start and when the dust settles and the gifts are opened make sure all the paper products go into the recycling bin. Simple enough, right? OR consider skipping the paper all together and wrapping your gifts in fabric wrap.
Option #5 : Use what you already have. My house has a big stack of maps thanks to my dad's subscription to National Geographic. Maps on their own don't say holiday but when stacked under a tree still look fun. What do you have at your house that could be turned into gift wrap? Tie it with string and be sure it gets reused or recycled once it's been opened.
Option #6 : Plan out your gift giving strategy. I work in retail and understand just how much stuff there is to buy. It's overwhelming. The holidays are overwhelming. Make notes and refuse to over-buy. Giving gifts is wonderful. Giving gifts that people will use and enjoy, not just discard, is the key.