Chris, congratulations! Seemingly rare Y-STR repeats can be very useful, genealogically. I say "seemingly" because every year--almost every month, it feels like--we add more and more data to the catalog of information we have about the human genome. One ramification is that some mutations which once seemed exceedingly rare are turning out to be less so as more and more data are added. The great resource that Peter linked to, the collection of Y-STR frequencies compiled by Leo Little, is unfortunately over a decade old. Leo was a fellow Texan and, unfortunately, I met him only once; he passed away in May 2008.
Leo worked with 67 Y-STR markers, state-of-the-art at the time. Now FTDNA tests for up to 500, and we have over one million STRs throughout the genome classified and cataloged. ISOGG has been doing a stellar job the past several months reacting to the flood of newly-identified Y-SNPs coming in thanks to rapid growth in Y-chromosome full sequence testing, and they currently have just over 77,000 haplogroup nodes, or indicative SNPs, in their catalog. ISOGG also notes that the number represents less than 20% of the identified SNPs; the other 310,000 await cataloging and we'll no doubt see more identified before they're entered. Sites like Yfull and Alex Williamson's The Big Tree index more Y-SNPs than does ISOGG right now.
Subsequent to Leo's work, more detail was uncovered about the average mutation rate of DYS426...and it's a slow mover. Ravid-Amir and Rosset (Bioinformatics. 2010 Sep 15; 26(18): i440–i445) worked it out to be a per-generation mutation rate of about 7.5 × 10−5 (± 3.7 × 10−5), so call it a 0.007% chance of a mutation taking place in any given generation. Ain't breakin' any land world speed records.
You didn't tell us anything else about your results, but I glanced at the West surname project at FTDNA at https://www.familytreedna.com/groups/westdna/. Check the numbers in the very first grouping: https://www.familytreedna.com/public/WESTDNA?iframe=yresults. The project admin has the group labeled as Family Group 01, Robert West, 1612, England. A whole bunch of M269 men with DYS426=14.
The only one in the group who shows (publicly) as having taken a SNP test for deep subclades is the one displaying the R-BY27945 haplogroup. That could be consistent with the possibility that the DYS426 STR may have mutated from the more common 12 repeats to 13 repeats first, and then to 14 repeats...all a lot of generations ago.
STR values are never perfect at predicting haplogroups, but they can offer a guesstimate. We learned a few years ago that the combination of DYS393=13 (which all the men in that group are, and I'll bet you are, too) plus DYS426=13 is indicative of L51+ (aka M412) P311-, an intermediary subclade between L23 and L11, found today with greatest density in Central Europe, but found from Iran to England. DYS426=13 isn't the most common value in the world, either.
If you take the long walk down the R haplogroup clades, you'll see: M269 > L23 > L51 > P310 > L151 > P312 > DF27 > Z195 > Z198 > ZS312 > BY3255 > BY27944/BY27945.
A couple of thoughts on moving forward. First, go to the West DNA project and compare your STR values to those in that FG 01 group. If they look the same, or at least mostly the same (which I kinda suspect they will), a lot of the kits that no doubt show as being matches in your FTDNA account are probably there already. If so, join that project. If you are an exact 67-marker match with any of those Wests, I'll hazard you've found a yDNA home to work with.
If you have another paternal-line relative to test, like a sibling or uncle or known male-line 1st cousin, you can encourage him to take a Y-37 test at FTDNA. If money is an issue (though there's a nifty sale on now through the end of August), he--or even you, if you want 3rd-party confirmation--can have DYS426 tested as a single STR at YSEQ (https://yseq.net/) for $9.95 plus shipping. A male relative also showing 14 repeats would rule out any possible testing/interpretation error in your results.
Upping the game to a deep SNP panel or the Big Y-500 test will tell you more, but if you are a solid match to our new BY27945 friend, you can order a single SNP test from FTDNA or YSEQ. Single Y-SNPs are not the best option because it is possible to test positive for one deeper SNP along the hierarchy while otherwise testing negative for the others upstream. But it's a cheaper way of making an educated guess. If you do, I wouldn't go for BY27945 even if it's available...which it probably isn't. I'd be more conservative and look at ZS312 or, a step higher up the ladder still, Z198. Z198 is available from YSEQ as a single for $18.00, and should be available to you as an upgrade at FTDNA while viewing your yDNA haplotree.
If all those particular stars align, it seems a reasonable chance that your surname may have derived as a patronymic, e.g., "son of West" or similar, and that the group of West descendants in the DNA project's FG 01 group represent a shared common paternal ancestor several to many hundreds of years ago. I could be wildly stretching things from you mentioning just one STR marker, but there could be something there to chase.
Edited: to change a lost-significant-digit mistake and correct 0.07% to 0.007%. :-/