The main issue with Alexa is the privacy concern. Users are not generally happy about the fact that all their conversations are recorded and stored. Amazon responded to this by saying that not all conversations are restored and the device only starts recording when it hears the wake word.
Solution: Amazon can let users have complete control over the recordings of their conversations.
Even though the word Alexa seems very short and easy to pronounce, there are glitches in the activation of the device based on how it is pronounced. This could be a big problem for users in foreign countries. Apart from that, the activation of Echo depends on how loudly the wake word is said. I have personally encountered situations when the wake word was being shouted out, but Echo was still not responding. This skit from Saturday Night Live frames this problem hilariously.
Solution: Amazon can let users pick their own wake words and customize it. This will not only make the experience more fun, but also differentiate each user's device from another.
You can connect multiple Alexa to multiple applications. One of them is Spotify, which you can use on multiple devices. When you ask Alexa to play a track on Spotify, if another user is listening to music using the same Spotify account but on a different device, Alexa will overrule the app and Spotify will stop playing music on the other device. This can really annoying. I personally use my Spotify account on my phone quite a lot, and when my roommates try to play something and request Alexa to do it, it cuts the music on my phone.
Solution: Amazon Echo can inform users that another user is using the app on a different device. Then ask them if they really want to stop using the app on the other device and use it on Echo instead.