Here are the two best recipes for electrically conductive adhesives I discovered a while ago, which can be cheaply made at home. I've been meaning to put these online for ages. Both of these solder glues will harden rigid, not flexible.
Please keep in mind there is no solder glue that performs as well as actual soldering (or spot welding). Since there are only a few things like copper and steel that will solder easily, these conductive glues are best used where soldering is inconvenient or exceptionally hard to do. Conductive glue could also make a fairly decent fusible link. The drawbacks are a higher resistance and breaking loose with much less force than actual soldering.
The number one combination I've found is lead-free solder powder with wood glue. It'll harden very rapidly when the two are mixed (under a minute), but a drop the size of a pencil eraser can get well below 100 ohms of resistance (I've achieved less than 10 ohms in practice). It can bond conductively to the bare surface of solar cells. I've tested both Tite Bond II original/weatherproof and 'silver' plumbing solder filed down into a powder (don't use rosin core if you take the same approach). Silver solder is actually about 10% silver or less, and I bet any kind of solder powder will work here. Solder powder containing lead is a health hazard, so I don't recommend using that. Any wood glues that are PVA based should work (yellow colored ones). Carbon and graphite powders don't seem to work as a substitute for powdered solder when combined with wood glue.
The next best conductive adhesive is spar varnish mixed with powdered graphite, carbon, or solder (any conductive powder). Typically, a glue that works with metal powders usually doesn't work with activated carbon and graphite, but spar varnish is the exception. It's the most versatile, or forgiving depending on how you look at it. It doesn't seem to conduct as well as wood glue and metal powder can (it varies, but is usually over 100 ohms). Using a metal powder conducts better than carbon or graphite (with the volume of each powder being equal). Spar varnish is also known as outdoor or marine varnish. It can be bought in small cans at any general hardware store. Being transparent makes it more vulnerable to UV breakdown in sunlight, which could be a drawback.
Because both of these mixtures harden rapidly, it's a good idea to mix up the smallest amounts possible and use them immediately. If your mix hardens the moment the glue and powder are combined, you're probably trying to use too much powder. As a rule of thumb, it's better to add powder to glue, not glue to powder. Toothpicks are good for mixing and applying. Sources of heat nearby may speed up hardening and work against you (like working in direct sunlight).
Whatever materials are being bonded, their surfaces need to be clean and oil free. If it's possible to rough up the surface of the materials being bonded with sandpaper, that'll also help things stick together (but I don't recommend roughing up the surface of a solar cell). If the bond isn't strong enough on its own and the stuff keeps coming off, try putting a thin layer of epoxy or superglue over the conductive glue, or use a mechanical fastener (i.e. twist tie, clamp, etc). Be careful with superglue though, that stuff can ignite if you mix it directly with any powder.
To get consistent results mixing glues and powders, I recommend building a mixing board. At its simplest, this is a block of wood with two metal sheets fastened on it. The sheets are connected with alligator clips to a resistance meter, and a narrow gap between the sheets separates them. When the powder and glue are being mixed together on top of either sheet, spreading a bit of it on the gap between the metal sheets shows how much resistance the solder glue will have. Keep in mind the resistance will go down a bit more after the glue hardens. It's better to use actual sheets of metal, because foil will tear very easily. When I built mine, I used vent duct metal screwed onto a two by four block of wood (vent duct metal can be cut with a pair of strong scissors, utility knife or tin snips). If you build one like I did, blunt the edges and corners with coarse sandpaper or a metal file because they'll be sharp. The gap between the mixing board sheets will need to be cleaned out after every use. After using it several times, the metal sheets will probably need to be sanded or scraped clean.
There are a lot of other glue and powder combinations that can also work, to an extent. For example, latex or acrylic paint mixed with powdered activated carbon or graphite. Be aware not all mixtures will conductively bond or stick as well to every kind of material. It seems the majority of conductive adhesives that can be bought pre-made are epoxy based, but in my own tests I was never able to get JB Weld, Bondo or fiberglass resin to be conductive (epoxy seems to insulate too well).
Disclaimer: Avoid inhaling or ingesting any metal powders - they're most likely harmful even if they don't contain lead. I haven't tested anything above 12 volts and 0.25 watts, and using any solder glue with higher voltages or currents might be a fire or shock hazard. Don't mix superglue with powders (or cotton) because it may get very hot and ignite. Connecting different metals together might cause galvanic reactions which rapidly corrode.