Both choices seem acceptable.
I'd tend towards the RJ11 solution and use 6P6C for flexibility unless you were absolutely certain no extra facilities would be needed subsequently.
Both systems run the risk of other equipment being plugged in by careless users in an uncontrolled environment.
Cannon a.k.a. XLR connector (see below) is a superb solution if you need the robustness.
Stereo audio cable has the advantage of shielded balanced pairs being available if desired. While I'd expect a 10 metre cable length to be no problem in a benign environment, the shielding is a bonus in severe environments. Balanced pair is less of an issue as you will be using an unbalanced against ground data circuit. Reasonable cost. Cables are available that stand up well to use as a flexible lead. 4 pole versions are available. Connectors tend to short momentarily on insertion and removal - not usually an issue but worth noting. Some risk of connector damage from ham fisted users.
"RJ11" 6P4C or 6P6C cable and connectors has the advantages of low cost, good availability, balanced pairs, additional pair for "something else that turned out to be handy" if 6P6C is used. (Even 6P4C gives an extra wire (unbalanced against ground) for "other use". Cable is designed for use as a flexible cord within reason so is damage resistant. Current carrying capability is reasonable (not a major issue here). Shielding not usually provided. Locking tab connectors. Cheap versions prone to lock tab damage but otherwise resistant to user abuse. Versions can be had which allow pullout without damage - cheaper versions tend to break locking tabs off if abused. Not so good for inline connections usually (but parts are available if this is required).
Other:
Nothing stands out except perhaps Cannon microphone connectors.
"Cannon" a.k.a. XLR*: Possibly the most professional solution at a reasonable price if robustness matters. 3 pin and ground and higher pin count versions. Larger connectors than RJ11 or 3.5mm. Connectors available from cheap plastic with modest robustness to reasonably priced near bulletproof full-metal-jacket versions. Latchable. Can use variations on shielded or unshielded cable
Many versions
6.5mm microphone connectors. Very available. Good prices. Larger connectors. Various number of poles. More user damage resistant than 3.5mm. (The photo below actually shows adaptors but the plug barrels are the same.)
Video cable with DB15 connectors. Thicker, more expensive. Available premade in various lengths to beyond 10m. Some shielded pairs or whole cable may be shielded. Quality connector. Larger connectors. Not usually good for inline connection.
Mini-DIN and similar: Small. Dearer. Damage prone. More connections but no real advantage.
Std DIN. Larger. Less damage prone. More connections but no real advantage.
Cannon / XLR:
"The XLR connector was invented by James H. Cannon, founder of Cannon Electric in Los Angeles, California (now part of ITT Corporation), and for this reason it is sometimes colloquially known as a cannon plug or cannon connector.
Originally manufactured as the Cannon X series, subsequent versions added a latch (Cannon XL) and then surrounded the female contacts with a resilient polychloroprene, which resulted in the part number prefix XLR
Source - Wikipedia.
It is a long question, but better than a short one, as you've shown your own research.
1) Solar cells. If you're stacking your own ones, stack 9 of them and get the 4.5V of the original circuit.
2) Battery charging. Batteries are the only thing you've left out of your spec. This is an area where the circuit design relies on cutting a lot of corners. In theory it might be out of spec, if you were to put 4.5V at 280ma through AA NiMH cells indefinitely. In practice, you don't get full sun all day, you'll be using it indoors, and you're not going to get optimal power transfer from the cells, so this isn't going to cause problems.
3) Diode. It's just a regular diode, not a zener. Current through it is actually determined by the battery and right hand side circuit, not the solar panel - the transistor is off when the panel is generating electricity. The original 1N914 will be fine. 1N4004 will also be fine.
4) Resistors: not a precision component here, use whatever meets your cost constraint. 5.1k for 5k is fine.
5) Wire: not critical. Your ebay link looks suitable. Thinner is better for the toroid.
6) Transistors: stick with the exact part numbers. Design may rely on specific parameters.
7) LED: again, this circuit relies on cheating. Normally a white LED won't run from two NiMH cells. The joule thief part provides a boost converter that gives small pulses of higher voltage. It doesn't have the capacity to provide a lot of current at that voltage. In combination with the pulsing this means there should be no risk of damaging it.
(A proper analysis of this circuit would be good, if nobody else supplies one I'll do it in a few days).
Best Answer
The conservative NEC set the maximum current density of about 700A/cm^2 for normal house electrical installation with common gauges copper wire. With AWG22 even at 1A the current density it is half of this limit (about 300A/cm^2).
However the PVC insulation of the telephone cable is terrible, because their purpose is other than current feed line use. But even the AWG22 bare wire, temperature rise is only 10 degr Celcius if carry 1A.
The voltage drop of a 30 ft long wire AWG22 carrying 1A at 24 DC voltage is about 1V or 4%, with a maximum allowable drop of 5%.
All above valid for straight cable suspended in air at a normal environment