OK, let's get this answered...
First of all, let's put aside the issue of whether you can use "architect" as a verb. In this job title, "architect" is definitely being used as a noun.
In contrast, "solution" or "solutions" is a noun modifier or noun adjunct. A noun adjunct is a noun being used to modify the sense of another noun and it fills a similar role to that of an adjective. You can see this by comparing some similar constructs:
- residential architect
- landscape architect
- software architect
- solution architect
- solutions architect
The first two job titles refer to specialties within traditional architecture, the first using the adjective "residential" and the second the noun adjunct "landscape". Items 3-5 are all IT job titles and all use noun adjuncts. Because "software" is an uncountable, mass noun, there's no option to choose between singular and plural forms. But when we use count nouns as adjuncts, we have to decide whether to use the singular or plural form.
The standard grammatical approach used to be to use the singular form as a noun adjunct unless this would change the meaning. So we have "market trader" and "stock trader" but "futures trader" and "arms trader". Why? Because the latter pair do not make their money from selling the future or upper extremities.
With the profusion of new and co-opted vocabulary that the IT industry has created there are many more occasions now when people are using previously countable nouns (e.g. "solution", "operation") in plural form for a generic field of endeavor. Thus, the practice of keeping systems running becomes "operations" and we don't really think about each individual singular "operation" that makes this up.
For this reason it's natural to use the plural version of the noun when forming "operations manager".
The concept of "solutions" is part way along this scale. We do talk about a specific "solution" in the sense of a product or configuration. But this type of "solution" isn't very countable -- you could imagine choosing among three solutions, but you probably wouldn't say you designed 171 solutions in May. In many cases, IT "solutions" are not discrete items but just the general concept of an implemented and integrated set of products.
So, when it comes to picking a job title, people have trouble deciding which of their gut instincts to trust and you end up with both solutions architects and solution architects.
My own feeling is that solutions architect is the more common. My experience is that almost no-one else cares which form I use.
Source: I am a solution(s) architect and a former copy editor