I want to buy a hat from merchant A for 0.0001 btc. I head over to my Electrum Android mobile wallet and open a Lightning channel for the first time. What's the minimum amount of btc I should use to open this channel? I suppose it should be the cost of the hat (0.0001 btc) + the mining fees, correct? If so, then at current mining fees of 0.0001088 btc, the transaction fees will be higher than the price of the hat.
The default minimum changes with different wallets and nodes AFAIK, but you should have enough to open the channel and cover the air of what you will want to pay for and also have left over to cover the tx fees when you want to close the channel.
You should not just open a channel for one small transaction as you highlighted here.
But let's say I go ahead and open the channel anyway, and I do so for 0.0002088 (0.0001 cost of the hat + 0.0001088 mining fees to open the channel). I then send 0.0001088 to merchant A.
1 month later, I want to purchase a bracelet from the same merchant, would I have to open another channel?
No.
You should open the channel with enough bitcoins to cover other small transaction you will make over the next couple of months.