I apologize for the newbie question, but I want to send ETH from my Poloniex account to someone else's. Is it possible to do this directly from my Poloniex account?
Yes, it should be possible:
In your first account (from where you are sending), go to https://poloniex.com/balances, find ETH, click Withdraw.
Open your second account in a different browser, go to https://poloniex.com/balances, find ETH, click Deposit. Check the Deposit address and save it.
Go back to your first account, insert the deposit address from the second account.
You should try transferring a small amount first to check if it works.
Just in case someone else is not seeing a record of their transaction, here's the reply I received from Poloniex's support:
The minimum amount of ETH that can be deposited here before it shows in your account is 1. The next time you will deposit ETH and the sum of the two deposits is bigger then 1, the total amount including your last deposit will be credited.