Check the security of the cables - ask any AA man, or someone who regularly attends vehicle breakdowns - it's as often the battery connections as the battery - usually a loose or corroded negative earth strap.
Check the attachment points on the starter and where they bolt to the solenoid and/or frame. If the cables look to be in good condition, securely attached and uncorroded, look elsewhere. The solenoid is clicking because there isn't enough power being sent to it to operate it correctly - the reason may be a bad battery, but it may be a good battery, with poor cables unable to transfer its power to the solenoid...
All text & original photos © Mike Oughton 2004 - 2015