You think that's strange? I've seen minor races build mining starbases to harvest the influence adding resource.

(I should have posted this earlier, but I forgot and reading your post caused me to remember.)

Now that I think about it, maybe there should be a specific AI for minor races only.

For example, perhaps minor race AI's won't value as much influence related techs a player is offering in a trade, since the only value they have to the minor race is to offer in trade to a third party.

Not wholely true - they do accumulate influence, and will often dump influence starbases in other AOIs, but the latter usually - not always - have no modules attached so are useless anyway and only serve to extend range/annoy.

Its often useful to take out a minor race as part of a takeover strategy of another Race as the minors accumulate substantial influence points to prevent cultural takeover. When you grab them its like a Culture Bomb that rips apart the opposing Race AOI. A recent game I took a minor race with 204,000 influence points, on grabbing their planet, and paying cash for a political Capital (they are very cheap), the whole AOI of the surrounding race was blown apart and planets started to revolt.

Regards
Zy