You'll want to use a cable not individual wires here
What you will want to do is lash a cable, specifically a UF cable (the same one you'd use for direct burying the run to the sign) to the messenger, not individual wires. Your initial plan is a Code violation outside of an industrial establishment where only qualified personnel service the installation, even, due to NEC 300.3(A):
(A) Single Conductors. Single conductors specified in Table
310.104(A) shall only be installed where part of a recognized
wiring method of Chapter 3.
and NEC 396.10:
396.10 Uses Permitted.
(A) Cable Types. The cable types in Table 396.10(A) shall be
permitted to be installed in messenger-supported wiring under
the conditions described in the article or section referenced
(B) In Industrial Establishments. In industrial establishments
only, where conditions of maintenance and supervision ensure
that only qualified persons service the installed messenger-supported wiring, the following shall be permitted:
(1) Any of the conductor types shown in Table 310.104(A) or
(2) MV cable
Where exposed to weather, conductors shall be listed for use
in wet locations. Where exposed to direct rays of the sun,
conductors or cables shall be sunlight resistant.
As to that messenger wire...
You will need to use a sag and tension chart, or run the NESC sag and tension calculations (see PDF pages 33-35 of this Southwire manual for details) yourself, in order to figure out what size and tension of messenger is acceptable, given the clearance requirements imposed upon you by NEC 225.18:
225.18 Clearance for Overhead Conductors and Cables.
Overhead spans of open conductors and open multiconductor
cables of not over 1000 volts, nominal, shall have a clearance of
not less than the following:
(1) 3.0 m (10 ft) — above finished grade, sidewalks, or from
any platform or projection that will permit personal
contact where the voltage does not exceed 150 volts to
ground and accessible to pedestrians only
(2) 3.7 m (12 ft) — over residential property and driveways,
and those commercial areas not subject to truck traffic
where the voltage does not exceed 300 volts to ground
(3) 4.5 m (15 ft) — for those areas listed in the 3.7 m (12 ft)
classification where the voltage exceeds 300 volts to
(4) 5.5 m (18 ft) — over public streets, alleys, roads, parking
areas subject to truck traffic, driveways on other than residential property, and other land traversed by vehicles,
such as cultivated, grazing, forest, and orchard
(5) 7.5 m (24 1∕2 ft) — over track rails of railroads
In general, this messenger will be a steel or ACSR wire, and it must be grounded to the circuit ground wire, although this can happen at either or both ends. (This is simply a matter of using an appropriately sized split bolt and 12AWG bare copper jumper to wire the messenger to the mains junction point, where it can connect to the circuit grounding wire.)
The messenger's size will need to be determined by the NESC tension calculations -- it's a bit of an interactive process that depends on where you are and how much sag you can work with, as the mechanical aspects control here, not the ampacity of the wire.
...and the lashings
There are two basic approaches to attaching wires to a messenger -- you can either use a lashing wire that wraps around the messenger and carried cable, or you can use cable saddles or bridle rings that clamp onto the messenger and provide an ring the carried cable is run through to support it. The latter is probably easier for someone who isn't running overhead cable for a living, albeit somewhat harder to find (you'll need to ask your local electrical supply house for them, but you'll be shopping there anyway, right?).
Furthermore, you'll need to remember to put drip loops into the ends of the UF so that water can't run down the UF into the junctions or mastheads at each end.