It's not true that random sequences of letters necessarily do not have meaning. I need simply produce a simple counterexample. If meaning is understood as association between sequences of graphemes and human experience, the simplicity of the binary representation that encodes the meaning is in no way tied to meaning. That being said, you can create any theory you want. Ultimately, you'll need a community to read, argue over, and adopt your theory as an explanatory tool.
One can design a cipher so that one maps meaning indirectly to a set S of n randomly generated sequences. Now, let's say the set looks like:
ant asleep at the wheel = alsegn
bee flying at a frightening pace through the world = febllp
cat sitting on a really, really, really hot tin roof = quzifes
all words in the OED starting with a letter = ggmoxf
Now, each random sequence of has cryptographic utility as well as meaning. Thus, the mapping of meaning to, let's say binary sequences that are used to encode the alphabet of any collection of graphemes has no relationship of proportionality between length and the complexity of meaning.
But let's roll your idea back to just standard English vocabulary. What does it even mean for meaning to be quantified? Does semantics even give us any leads? Do philosophers of language have anything corresponds? If one takes a pictures of something meaningful to a human, are the configuration of pixels somehow indicative of the complexity of the image? How about the meaning of the picture? Well, there is the principle of compositionality. From WP
The principle of compositionality states that in a meaningful sentence, if the lexical parts are taken out of the sentence, what remains will be the rules of composition. Take, for example, the sentence "Socrates was a man". Once the meaningful lexical items are taken away—"Socrates" and "man"—what is left is the pseudo-sentence, "S was a M". The task becomes a matter of describing what the connection is between S and M.
So, certainly in natural language, the complexity of meaning seems to enjoy a "rough" proportionality between morphemes and amount of meaning. For instance, "myocardial infarction" seems to have more meaning than "heart" intuitively. And "I suffered from myocardial infraction last day of last year" has even more meaning there. But in some ways, it counts on how you measure meaning. For instance, can you say with certainty that "borborygmus" has more meaning than "fart" despite one has more syllables than the other? Is their conceptual semantic content not almost identical? In fact, besides multiple theories of semantics, philosophy recognizes that meaning has all sorts of dimensions. Which is meaning? Again from WP:
The major contemporary positions of meaning come under the following partial definitions of meaning:
Psychological theories, involving notions of thought, intention, or understanding;
Logical theories, involving notions such as intension, cognitive content, or sense, along with extension, reference, or denotation;
Message, content, information, or communication;
Usage, and the instructions for usage; and
Measurement, computation, or operation.
In conclusion, Shannon's theory is a great leap forward in quantifying strings of symbols, and has philosophical import in measuring simplicity and complexity, doing hashes for algorithmic efficiency, doing cryptographic work, etc., but to impute some sort of essence on physically real objects based on their natural language representations is dubious for reasons not the least of which that strictly speaking syllabaries are artifacts of human language communities, not aspects of objects. Hence, to do so would confuse the territory for the map; in other words, would mistake the symbol for the referent.
So, from a naturalized epistemology, the theory of information is best understood as a mathematical tool for analyzing languages or sign-systems invented by people, where the latter are abstractions.
Is meaning an intrinsic property of objects (as is Shannon's quantity of information) or is it something entirely subjective that therefore does not admit of objective quantification.
Meaning is probably best understood simply as an experience of agents and might be understood in terms of mental representations.
And, if we somehow restrict the possible images or sequences, then for that set a quantitative measure of its meaning could be constructed.
Sure, one certainly can measure the language used to convey meaning, but is that the same as measuring meaning itself? Perhaps with fMRIs measuring NCCs?
On the whole, you're free to create whatever theory of meaning pleases you. But ultimately such a theory will be judged on its explanatory power or utility by your fellow thinker, and there's a fine line between crank and genius.