Could it be the solenoid on the starter?

Sounds like a possibility. If you have a meter or test light it's fairly easy to check, but you need a helper:

1) Get access and visibility to the smaller wire at the solenoid (push-on terminal),

2) Leaving the wire attached, probe for +12 volts when helper turns key to the Start position (be prepared so you won't be startled if it should start).

3) If you get +12 v but no cranking (and no solenoid click), the solenoid is bad.

4) If it clicks but doesn't crank, its the starter.

Either 3 or 4 may be repairable. Hope this helps,
--
Bruce Young,
'93 940-NA (current)
'80 GLE V8 (Now gone)
'83 Turbo 245
'73 142 (98K)
'71 144 (track modified--and still here)
New 144 from '67 to '78
Used '62 122 from '63 to '67