My advice would be to forget about fleshing out a sophisticated system until you've demonstrated solid execution and profitability on the business end of things. Until you have more than a few hundred customers, it would be financially irresponsible to start involving computer scientists in a process that you could just as easily handle yourself with a spreadsheet.
So, here's what I would do as a non-technical person trying to start this up:
* Figure out what you want as far as branding, draft up all your content and marketing materials, think about what you want out of a landing page and sketch out some basic wireframes, etc.
* Make your survey in SurveyMonkey, Qualtrics, or a similar service.
* Invest in a good logo ($300 to $500). 99designs seems like a pretty nice service for this: http://99designs.com/logo-design
* Find a good developer/designer to make your site, populate it with your content, integrate it with Stripe for recurring payments (with an optional field for them to increase their monthly gift budget), integrate your survey (in an iframe or something), and throw together a very basic database-driven backend for storing login credentials and customer information (including the link to their survey results). At this stage, the backend should also shoot you an email each time you get a new signup. I'd budget $1000 to $1500 for the site.
* Each time you get a new signup, send that person a personalised email introducing yourself, thanking them for signing up, prying more into their survey answers for more specific information, and specifically making yourself available if they ever in the future have questions, feedback, or updates about their SOs which they'd wish to share (insofar gift-selection is concerned, of course).
* Once you have all the information you need about the customer's SO, make an entry in a spreadsheet containing their name and address, as well as a general category of gifts they'd be into and a couple specific notes about what they like. Ideally, the general categories would be broad enough to be reusable between customers, while the specific details would be focused interests (certain films/shows/music, specific interests, etc.). If you want to be really thorough, you could even in your dialogue offer to look at a copy of the customer's SO's entire trove of Facebook "Likes".
* Now, the fun part. Each month, you get to look at your spreadsheet, verify which customers are still paying, email the ones who've left soliciting feedback, and pick a specific gift for each one who hasn't. As far as gifts, I'd mostly stick to Amazon, since it tends to be reliable and well-priced, it aligns with your goal to have the shipment "just look like he purchased something off of Amazon", and Amazon already offers the option to gift wrap for an extra fee (no point in getting your hands dirty with wrapping/packaging/shipping when you can just dropship at a much lower opportunity cost). As a clever scalability hack, rather than have a flurry of gift-shopping at the same time each month, schedule each customer's gift shipment for a different day of the month (i.e. calculate the modulo of each customer's ID number in the database with 28 plus 1 and ship the gift on that date).
* If you want a nice starting point for choosing gifts, http://dowant.net/ is a little-known gem with a semi-frequently updated list of cool and quirky gift ideas from Amazon. (Speaking of which, the owner of Do Want is a pretty good and well-priced development/design/artwork freelancer if you want his contact info.)
* Aside from the day-to-day stuff, marketing still applies. Look at Google ads, Facebook ads, Reddit ads, a "Show HN" post, reddit.com/r/shutupandtakemymoney, paying for Twitter backlinks on Fiverr, etc. Hell, if you execute well I could even consider plugging it on the Relationship Advice subreddit (which I created a number of years ago and still moderate). Aside from the usual online stuff, I wouldn't be quick to discount the value of geographically targeted physical ads; see if you can look into cheap adspace in locations known to be relationship/honeymoon retreats, for example.
There: for less than $2000 (plus minimal recurring costs of domain+hosting and/or whatever you can budget for advertising) and a bit of legwork, you have a solid MVP for a fully running, attractive, self-sustaining business. If you can eke out just $10 mean profit per customer per month, meeting a goal of 100 paid signups would put you at well past breaking even pretty quickly. A few hundred more than that, and (depending on how much manual labour you want to put into this) you could have a full-time job on your hands.
Long-term, as soon as things are on track to conflict with your day job (assuming you'll not have not left by this point) and you have the marginal cash flow to justify it, you'll want to begin automating things and preparing to scale up:
* Obviously, you'll want to largely ditch the spreadsheet. The thing about automating what you want to do, is that automating it effectively and to a comparable quality of your manual operation is that it's actually a reasonably complex machine learning problem; don't expect to have it solved as quickly and cheaply as bashing out a Web site. At this point, I'd suggest finding a good CTO who'd be willing to work part-time for ~40% vested equity and take over the metamorphosis of your operations. This doesn't need to be her life at this stage (it won't have scaled to support her fully anyhow), but the benefit here is that you'll have someone to stick around and run the technological show once things really ramp up. She should have at least three to four months of runway to get an MVP ready for production and properly tested, so ideally you'd begin the search for a partner at a time when you're able to both demonstrate sufficient traction/profitability to attract quality talent and be able to continue manually handling all of the extra signups that you'd expect to receive over the next four months (if necessary, you could look into offloading some of this work to a high school intern or cheap contractor).
* As far as implementation, throwing together what you want completely from scratch isn't an easy problem to solve. However, a lot of very solid machine learning algorithms are already open source, and you'll already have M months of gift choice data matched up to N different broad gift choice categories, which are in turn already matched up to N different survey data sets. With all this data plus the Apache Mahout machine learning recommender engine, a lot of the work is already done for your CTO. You then only need to have your system scrape and process Amazon's data on your gifts to determine novel gift choices from Amazon for your customers. In addition, at this point, have your CTO add in an option for your users to link their accounts with Facebook, specifically requesting the "friends_likes" permission in the API (with an explicit explanation that the purpose of the optional Facebook link feature is to scrape a list of their their SO's interests for more focused gift choices); so, essentially, your new system now gets the benefits of the earlier "specific interests" on steroids.
* Until you have more than, say, a thousand users, and the new system is well-proven, you'll need to personally sign off on each gift choice before letting it get shipped out. You don't want any silly bugs tarnishing your reputation or accidentally buying gifts that will bankrupt you.
* Beyond this point, pretty much just keep iterating and sticking to it. (Or, if you lose interest, I suppose you could get it 100% automated then contract out support to India and move on to other things while your bank account grows.)
* As an aside, since Amazon doesn't allow you to buy things through your own affiliate links, cough, a convenient way to get a consultant/advisor available at all times for virtually free would be to make sure you and/or the automated system purchase every gift using that person's Amazon affiliate links.