Can aged PLA be refurbished?
The easiest way to freshen up filament is hot air, although there are other options.

There is an optimal melt processing moisture level for every plastic, typically in the range of 0.1-0.2% water content by weight. But the equilibrium moisture content of most plastics in humid air can be more like 1%. As a consequence, hot air dehydration is standard practice to prepare plastic pellets for extrusion in industry. Pretty much every injection-molded and extruded plastic product in the world -- including 3d printer filament -- is made with pre-dried pellets. Hot air is blown through the pellets until they are below the moisture limit.

Different plastics tolerate extrusion with different amounts of moisture, and absorb different amounts of moisture. They can also tolerate different drying temperatures. So the ideal storage and drying conditions vary by material. 


PLA absorbs relatively little moisture, but in humid environments can accumulate enough water content to cause steam bubbles during extrusion. In some cases, steam bubbles can contribute to hot end jamming. Extreme wetness has been known to cause swelling that can increase filament feed drag. It's arguable whether wetness actually causes brittleness, or if that is a separate aging issue. People who keep their homes below ~50% relative humidity usually don't have any problems. Leaving PLA in a dry environment for a week or two should adequately re-dry it, or it can be gently heated to about 120F / 50C for a couple hours. (Some people dry it hotter. but that risks deforming the filament.)
ABS also doesn't absorb very much water, but perhaps a little more than PLA. It also experiences steam bubbles, but that's typically the only issue. Homes below ~45% RH usually don't have problems. Storing ABS with fresh silica gel for a couple weeks will dry it. Or it can be oven-dried up to about 180F / 80C for an couple hours. HIPS can be treated the same. 
The kind of PET used in plastic bottles is rapidly degraded into tar by hydrolysis when melted with any significant water content. So PETG filaments are specially blended to absorb less water and to be less damaged by water than PET. But there is still the possibility of bubbles and cloudy strands due to steam expansion at higher temperatures, and there is some evidence that wet PETG produces weaker, more brittle prints than dry PETG. Different manufacturers' blends require different conditions: some people report Taulman t-glase needs oven-drying and careful storage while Esun PETG is fairly tolerant of normal home humidity levels. Oven-drying at 150F / 65C should work well. 
Nylon absorbs a huge amount of water, which causes it to swell considerably, produce massive steam bubbles, look cloudy, warp more, and adhere less than properly-dried nylon. It's nearly unprintable when wet. It should be oven-dried at around 150F / 65C for 4+ hours -- desiccant will not strip enough moisture from it. In fact, nylon will pull water out of used silica gel! Once dry, it should be stored with an aggressive desiccant (either bone-dry silica or preferably calcium chloride). It only takes a few hours of exposure to air for it to become excessively moist. Building a sealed drybox feed system is highly recommended to avoid exposed time during and between prints. 
Polycarbonate is similar to nylon in that is is an aggressive water-absorber. It will look cloudy, produce steam bubbles, warp more, and provide very poor layer bonding when wet. It should be oven-dried at 180F / 80C if on a plastic spool (up to 250F / 120C if dried alone) for 4+ hours and then stored in a drybox with aggressive desiccant just like nylon. Note that some modern PC blends like Esun ePC are less prone to water absorption, at the cost of some decrease in mechanical properties. 
PVA is basically destroyed by airborne humidity, since it literally dissolves in water. Store in a drybox with an aggressive desiccant at all times. 


Composite filaments should be treated like the base material. 

Some drying and prevention options:


Oven: The "warm" setting will usually work pretty well. Let the oven preheat and settle out for a while, and measure temperature with a good oven thermometer or thermocouple. Shield the filament from direct radiant heating and hot spots with aluminum foil, cookie sheets, etc. Electric ovens will dry faster than gas ovens, because burning natural gas produces some additional moisture. Do not leave the oven unattended if using temperatures above the glass point of the filament, or bad things may happen!
"Light bucket": A 5-gallon plastic bucket with an incandescent lightbulb inside is a pretty effective way to gently warm low-temp filament like PLA for drying or medium-term storage. Leave the lid slightly open if drying. 
Food dehydrator: Works great. Set temperatures as per the oven temps above. The main challenge is getting a large enough space inside for a filament spool. 
Desiccant: In order for desiccant to actively dry filament, it must be significantly more attractive to water than the filament is. And affinity for water is a function of how wet the material already is. That means dry desiccant can easily pull some water out of very wet filament, but wet desiccant can actually give water TO the filament! Rechargeable indicator desiccant (such as an Eva-Dry E-333 unit) is ideal. It's also important to have ENOUGH desiccant: silica gel can only absorb 10% of its weight in water at 20% RH. That means to pull 1% moisture content out of a wet 1kg spool, you would need to start with at least 100g of bone-dry, fresh-baked silica! "Used" silica is basically useless, it already contains too much water to pull any more from the filament. But you can re-dry the silica in an oven. 250F / 120C for 6 hours should be safe for all types of silica gel, but more aggressive drying (including microwave drying) is possible for some silica gels. Follow the gel manufacturer instructions. 
Rice: Does not work. It's basically a myth that rice has drying power. A bag of rice you buy at the store is already pretty close to moisture equilibrium with the air, so it has minimal capacity to pull water from filament (or a soaked iPhone, for that matter). If you dry the rice in an oven to drive out its water first, it will work to some degree, but silica gel is considerably more effective. 
Kitty litter: Silica gel style kitty litter is nearly identical to desiccant silica gel. Like rice, it is fairly close to saturated when you open the container, but can be dried in the oven to be a good cheap bulk desiccant. Clay type kitty litter is not as effective. Be careful of getting kitty litter dust all over your filament. 
Dry storage A thick plastic box with an airtight seal is preferable. Look for recycling code 2 (HDPE) or 5 (PP) on the box. 5 gallon buckets with sealing lids also work fine. Ziplocks and other thin plastic bags are better than nothing, but are permeable to water (yes, really) and can only be relied on as long as there is fresh desiccant in the bag. Acrylic/plexiglas dryboxes have been sold by various people, but acrylic is very permeable to water, so I don't recommend that option. 
Air conditioning: Simply keeping the air in your printing environment reasonably dry will protect PLA and ABS and other low-absorption filaments. It may help to buy a humidity monitor to get an understanding of your ambient humidity. 


Now, considering the original question here, it's important to note that moisture content is not the only way filament can age or be damaged. PLA in particular is prone to becoming brittle over time. There are different theories for why this occurs. One is gradual chemical aging because poly(lactic acid) simply is not a very stable polymer. Moisture could contribute to that aging process, but true chemical aging would be irreversible even if the filament is later dried. How much this occurs should depend on the specific polymer blend and storage conditions. 

Another theory for PLA aging is that the residual filament extrusion stresses (from being drawn down to the correct diameter and rapidly quenched in a water bath) are slowly creeping over time. Anyone who has placed a PLA part under heavy load for more than a few weeks will see PLA creep. It's a rather odd polymer in that it will "creep to failure" and crack at very low creep elongations rather than progressively deform in a ductile manner like most creep-prone materials. So if the PLA has significant stresses locked-in from the initial extrusion process (which is very common) it may be creeping into a more brittle arrangement of polymer molecules over time. That would explain the aging effect, and it would explain why "drying" sometimes rejuvenates the PLA: heating the filament near its glass point will allow the polymer molecules to gently relax and basically anneal to a less brittle state.