Ok, the first thing to do is draw yourself a diagram and tag where the wires
are now, then take all connections loose from the solenoid. Set your meter
to read ohms, then check the following:
You should have continuity from the 'S' (left) small terminal to the 'I'
(right) small terminal - don't have exact figures, but probably from 5 to 15
ohms. This is the solenoid coil - the part that makes things move inside the
solenoid when DC voltage is applied across it. You should NOT have
continuity from either small terminal to ground (which would mean it's a
Ford starter solenoid), or from either small terminal to the right (battery)
large terminal (which would mean it's an Onan solenoid, but the wrong one
for your set).

The part number of the correct solenoid is 307-1617, probably about 40-45 at
your friendly local Onan dealer.

Once you have the correct solenoid in place, the following are the correct
hookups:

Left large terminal - one LARGE wire that goes into the generator end. One
smaller wire that goes to R1, a 150 ohm 5 watt resistor in the control box
(going to look kinda like a tootsie roll with two terminals, mounted
vertically on a screw in the bottom of the control box). One smaller wire
that goes to E1, the electric choke mechanism on the carburetor.

Right large terminal - one Large wire going to the battery +. One small wire
with a diode (CR1) in line, going to R2, the charge resistor mounted in the
generator end. This diode will be a flat chip, about the size of a chiclet,
with 2 spade terminals coming out of opposite sides, and normally would be
inside some sleeving. One small wire going to a terminal on F2, the right
hand (ignition) fuse on the control panel.

Left small terminal - one small wire going to terminal 3 (the left-hand
terminal) of the start/stop switch. One small wire going to terminal 85 of
K2, the left-hand relay in the bottom of the control box.

Right small terminal - one small wire going to terminal 87A of K3, the
right-hand relay in the bottom of the control box.

Some general thoughts: Onan is very good about labeling wires - all of your
original wires should be marked, assuming the markings haven't been worn off
or painted over. The markings are -printed- on the insulation, near the
connection, and identify what -this- end is connected to as well as what
the -other- end of the wire should be hooked to. For example, the wire from
the right small terminal should have K1-I / K3-87A printed on the insulation
near the starter solenoid, and K3-87A / K1-I printed on the wire near the K3
relay.  The wire from the left large terminal (SI) going to the electric
choke should be labeled K1-SI / E1 near the solenoid, and E1 / K1-SI near
the choke.

The exciter-cranked sets, like this one, require large battery cables, a
large batttery in good condition, and good connections to crank - since the
starter is part of the generator, it doesn't have any gear reduction to
allow the starter to spin at a high rpm. It's also possible that, if the
genset has been sitting for a long time, you have tarnish built up on the
commutator bars and brush tips - which will reduce the available cranking
power. And - most importantly - since the AC windings and DC windings are
both on the same rotor, a short on the AC output will make it almost
impossible to crank. I suggest that you first try cranking the genset with
the spark plugs out, after making SURE that the AC windings are disconnected
from the output wiring and insulated so they don't short (in the top rear of
the control box - the partition will slip up and remove after you loosen the
two bolts holding it). If it cranks ok, then hook up the AC output leads and
try again. If it's still ok, then put the spark plugs back in and see if you
can get it started.

Just for future reference, if you run into problems - is the genset
currently mounted in an rv, or do you have it out where you can get to
things?

Alan