I think that “on his nail” is a calque of the French “sur son ongle” which was a favourite phrase of the poet Jacques Carpentier de Marigny (1615–1673), according to the memoirs of his friend Gilles Ménage (1613–1692), in which it is explained as follows.
Lors que M. de Marigny† vouloit me marquer que j’étois fort de ses amis il me disoit que j’étois sur son ongle. C’étoit une façon de parler qui lui étoit ordinaire avec ses amis, il s’en servoit aussi dans ses lettres, & je vais vous en faire lire une qu’il m’écrivit de Flandres, où étoit alors. La voici: Ah, illustre de mon ongle ! […] Quand Marigny disoit à quelqu’un : vout êtes sur mon ongle, il donnoit à entendre deux choses ; l’une que cet homme-là lui étoit toujours présent, rien n’étant plus aisé que de regarder son ongle toutes les fois qu’on veut ; l’autre que les bons & les vrais amis sont si rares que l’homme du monde qui en auroit le plus n’auroit pas de
peine à mettre leurs noms sur son ongle.
When Marigny† wanted to show me that I was one of his best friends, he said that I was on his nail. This was a common way he had of speaking with his friends; he also used it in his letters, and I will have you read one that he wrote to me from Flanders, where he was then. Here it is: Ah, illustrious of my nail! […] When Marigny said to someone, you are on my nail, he meant two things: one, that this person was always present to him, nothing being more easy to look at, whenever he wanted, than his nail; the other, that good and true friends are so rare, that the man who had the most friends in the world, did not have so many that he could not write their names on his nail.
Gilles Ménage (1715). Menagiana, ou les bon mots et remarques critiques, historiques, morales, & d’érudition, de monsieur Ménage, recueillies par ses amis, 3rd edition, volume 3, pp. 193–194. Paris: Florentin Delaulne.
† There is a note in the 3rd edition that says that previous editions incorrectly attributed the phrase to Jean François Sarrazin rather than Marigny.
The first of these meanings is the one we want. To have something “on one’s nail” is to have it always present, nothing being easier to consult than one’s own fingernail. So in Aurora Leigh, Romney Leigh’s “pattern”, his plan for improving the lives of the poor, is ever-present in his mind, as Marigny said his friends were to him.
Is the Menagiana a plausible source? Well, we already have an example in Aurora Leigh of a calque from a French idiom, where “men in us” is a calque of “hommes en nous”, so it would not be too surprising if there was another. And Browning need not have picked it up directly from the Menagiana, as the anecdote, translated into English, appeared in A Dictionary of Literary Conversation (1795), p. 40, The Repository of Arts, Literature, Commerce, Manufactures, Fashions and Politics (1822), pp. 46–47, The Table Book (1827), col. 764, and no doubt elsewhere.
Marigny’s phrase seems to have remained idiosyncratic to him, and did not become an idiom in French. However, in Smollett’s translation of Don Quixote I find:
Quedó Sancho de nuevo como si jamás hubiera conocido a su señor, admirado de lo que sabía, pareciéndole que no debía de haber historia en el mundo ni suceso que no lo tuviese cifrado en la uña y clavado en la memoria
Sancho was, on this occasion, as much astonished at the learning of his master, as if he had never known him before that day; and imagined there was not an history or event in the whole world, that was not decyphered on his nail, or nailed to his memory.
Miguel de Cervantes (1615). Don Quixote, p. 525. Translated by Tobias Smollett (1792). London: Harrison & Co.
Here, “on his nail” seems to be an idiom for having something ready to hand. This seems a less likely source for Browning, however, as the meaning is not explained so clearly as in the Menagiana.