My first instinct is that this is a marketing question, and less directly a User Experience question. There are however some UX concerns with what you are asking that aren't exclusive of the marketing.
Branding: Is AndyXApp the brand you expect your users to find? Or is your current branding AndyApp where a search engine when queried with something like "Andy App to do Z' would return your site and possibly cause confusion? Especially when the following results are returned (and possibly others outranking your own site:
Security could be another concern, where random letters might look spammy to both the user AND spam filters. If this were and Android App for example, the user signs up for a service, through an App called "Andy". Will it be clear that AndyXApp.com or NoReply@AndyXApp.com is directly related to your site?
To wrap up, I'd avoid throwing a random letter in your domain just to get a domain name "close enough" to what you originally wanted.
1. As someone who has done sysadmin type work, it looks spammy and
could be flagged as such
2. From a marketing perspective it seems to dilute your brand
3. For the UX part of the question, it seems much
more likely to cause confusion than help your user
4. Your users might not remember where that random letter went! Was it
XAndyApp or AndyAppX? What's this AndyXApp.com, that doesn't look
right at all!
I'd suggest looking at what the site/ app does and look to build the brand around that: For example "Andy's App to show the user great travel deals that Andy finds while browsing various travel sites" could be "AndyDoesDeals.com" or perhaps a help site for your application: "HelpForAndyApp.com"
And, for the sake of the question, both of these domains are available right now!