Restoring an old home is really recycling on a grand scale. Extending the useful lifespan of a structure keeps literally tons of material out of landfills. Landfills themselves create an obvious environmental impact by altering the landscape. Additionally, much of demolition waste dumped in the landfill is wood (framing lumber, flooring, subfloor, sheathing, siding, lath, cabinets, etc.) and will decompose releasing carbon to the atmosphere. As long as wood is retained in a structure, it serves as a carbon sink for sequestered CO2.
Old homes are also typically in high density neighborhoods and hopefully near jobs, city services, and other urban amenities. This means less fossil fuel used in commuting compared to suburban and ex-urban dwellings. These old neighborhoods also use existing infrastructure (less tax expense, energy and resources required).
Most old homes are modestly sized. A small home used less material to be constructed (likely from local material), has a smaller literal footprint on the land, and requires less energy to heat and cool--a smaller carbon footprint (with a few energy saving retrofits). Features like useful porches not only provide for relaxation and interaction with others, but also work in lieu of air conditioning (enjoy the cool evening air giving the house time to cool down).
Double hung windows are taller than modern windows (nearly reaching the ceiling where the hot air is), with operable top sashes the warm air can escape at the top of the window and cooler air flows in at the bottom of the window to replace it. A double hung window can create an air flow loop that most modern window designs can not.
Lead based paint and asbestos have frightened some people away from old buildings, but if these things are sound, covered/encapsulated, and left undisturbed they are not a concern. Old materials are not off-gasing formaldehyde or VOC's and they are not made of supposedly "green products" shipped from China (that defeats the purpose). Wood floors in old homes last for generations, whereas the wall to wall carpeting in many newer homes goes to the landfill every decade or so. The Wood floors also harbor less dust and other allergens and are not made of petrochemicals.
Old homes get knocked for being drafty and poorly insulated. These issues can be addressed successfully with modest means. Old windows can be repaired and tuned up and with storm windows they perform well. Cellulose can be blown into wall cavities, attics insulated, etc. and tax credits or other programs may pay for much of this work.
Even if these building envelope upgrades can not produce a LEED certified home, the home may be greener than a certified new home due to the above reasons.
All the best,